Linear Combinations and Bases
In downtown Chicago you can give directions like this: "go 3 blocks east and 5 blocks north." You are really saying that you reach the spot by taking 3 steps of the vector $(1, 0)$ and 5 steps of the vector $(0, 1)$. Those two vectors are a basis. But they are not the only choice: what if a diagonal avenue cuts through the grid, and you want to give directions using that avenue and a cross street instead? In this page you will learn when a pair of vectors can describe the whole plane, and how to translate between different descriptions.
Many textbooks write vectors as $\langle 2, -3 \rangle$. On this page we write them as $(2, -3)$, the same way the exercises do.
Linear Combinations
You already know the two basic vector operations: multiplying by a scalar and adding. A linear combination just uses both at once.
Geometrically, $a\vec u$ stretches (or flips) $\vec u$, $b\vec v$ does the same to $\vec v$, and the parallelogram rule adds them tip to tail.
Often the unknowns are the scalars. Then you match components.
When a vector is drawn on a grid, read its components as run (horizontal change) and rise (vertical change). The components do not depend on where the vector starts.
This vector is $(3, 2)$. Scaling and combining vectors that are drawn works the same way: read the components, then compute.
Calculate the coordinates of the resulting vector of the linear combination $3\vec{a} - 2\vec{b}$, knowing that $\vec{a} = (1, 4)$ and $\vec{b} = (-2, 5)$.
Calculate the coordinates of the resulting vector of the linear combination $3\vec{a} - 2\vec{b}$, knowing that $\vec{a} = (1, 4)$ and $\vec{b} = (-2, 5)$.
Given that $\vec{w} = 2\vec{u} + k\vec{v}$, with $\vec{u} = (1, -1)$, $\vec{v} = (0, 3)$ and $\vec{w} = (2, 7)$, find the value of the scalar $k$.
Given that $\vec{w} = 2\vec{u} + k\vec{v}$, with $\vec{u} = (1, -1)$, $\vec{v} = (0, 3)$ and $\vec{w} = (2, 7)$, find the value of the scalar $k$.
The attached plane shows the vectors $\vec{u}$ and $\vec{v}$. Determine the coordinates in the standard basis of the resulting vector $\vec{z} = 2\vec{u} - \vec{v}$.
The attached plane shows the vectors $\vec{u}$ and $\vec{v}$. Determine the coordinates in the standard basis of the resulting vector $\vec{z} = 2\vec{u} - \vec{v}$.
Linear Dependence and Independence
Maya's example shows what can go wrong: if one vector is just a multiple of the other, the second adds nothing new.
Why the test works. If $\vec v = k\vec u$, then $u_1 v_2 - u_2 v_1 = u_1(ku_2) - u_2(ku_1) = 0$. When the components are proportional, the "cross products" cancel. (The expression $u_1 v_2 - u_2 v_1$ is the $2 \times 2$ determinant of the two vectors.)
Problems often hide an unknown in a vector and ask when the vectors are dependent. Set the expression equal to $0$.
If the parameter appears in several components, the equation may become quadratic. Then you get up to two values, and each one is a valid answer.
Given the vectors $\vec{u} = (2, -3)$ and $\vec{v} = (-4, 6)$, justify mathematically whether they can form a basis for the vector space $\mathbb{R}^2$.
Given the vectors $\vec{u} = (2, -3)$ and $\vec{v} = (-4, 6)$, justify mathematically whether they can form a basis for the vector space $\mathbb{R}^2$.
Determine the value of the parameter $m$ so that the vectors $\vec{p} = (m, 4)$ and $\vec{q} = (3, -2)$ are linearly dependent.
Determine the value of the parameter $m$ so that the vectors $\vec{p} = (m, 4)$ and $\vec{q} = (3, -2)$ are linearly dependent.
To guarantee the stability of a node in a planar lattice structure, the direction vectors of two concurrent bars must be linearly independent. The design vectors are…
To guarantee the stability of a node in a planar lattice structure, the direction vectors of two concurrent bars must be linearly independent. The design vectors are $\vec{d}_1 = (k+1, 3)$ and $\vec{d}_2 = (2, k)$. Determine for which values of the parameter $k$ the node is unstable (the vectors do not form a basis). Subsequently, assuming $k=0$, express the external load vector $\vec{P} = (4, -6)$ as a linear combination of $\vec{d}_1$ and $\vec{d}_2$.
Basis of the Plane and Coordinates
Why only two vectors? The plane has two independent directions. Two independent vectors are enough to reach every point, and each point is reached in only one way.
Drag the point: its coordinates tell you how many steps of $\vec\imath$ and of $\vec\jmath$ you take.
Express the vector $\vec{c} = (5, -2)$ as a linear combination of the standard basis vectors $\vec{i} = (1, 0)$ and $\vec{j} = (0, 1)$.
Express the vector $\vec{c} = (5, -2)$ as a linear combination of the standard basis vectors $\vec{i} = (1, 0)$ and $\vec{j} = (0, 1)$.
Reason whether three non-zero vectors in the plane, for example $\vec{u} = (1, 1)$, $\vec{v} = (-1, 2)$ and $\vec{w} = (3, 0)$, can be simultaneously linearly…
Reason whether three non-zero vectors in the plane, for example $\vec{u} = (1, 1)$, $\vec{v} = (-1, 2)$ and $\vec{w} = (3, 0)$, can be simultaneously linearly independent.
Orthogonal and Orthonormal Bases
Recall two tools. The dot product of $\vec u = (u_1, u_2)$ and $\vec v = (v_1, v_2)$ is $\vec u \cdot \vec v = u_1 v_1 + u_2 v_2$, and it equals $0$ exactly when the vectors are perpendicular. The magnitude of $\vec u$ (some problems call it the modulus) is $|\vec u| = \sqrt{u_1^2 + u_2^2}$. A vector with magnitude $1$ is a unit vector.
To build a vector perpendicular to $(a, b)$, swap the components and change one sign: $(-b, a)$. Every vector perpendicular to $(a, b)$ is a multiple of $(-b, a)$.
Check if the set of vectors $B = \{(-1, 2), (2, 1)\}$ forms an orthogonal basis for the plane.
Check if the set of vectors $B = \{(-1, 2), (2, 1)\}$ forms an orthogonal basis for the plane.
Check if the vectors $\vec{u} = (3, 0)$ and $\vec{v} = (0, -2)$ form an orthogonal basis for $\mathbb{R}^2$.
Check if the vectors $\vec{u} = (3, 0)$ and $\vec{v} = (0, -2)$ form an orthogonal basis for $\mathbb{R}^2$.
Determine a vector $\vec{v}$ such that the set $B = \{\vec{u}, \vec{v}\}$ forms an orthonormal basis, knowing that $\vec{u} = \left(\frac{3}{5}, \frac{-4}{5}\right)$. Is…
Determine a vector $\vec{v}$ such that the set $B = \{\vec{u}, \vec{v}\}$ forms an orthonormal basis, knowing that $\vec{u} = \left(\frac{3}{5}, \frac{-4}{5}\right)$. Is the solution unique? Justify your answer.
From Coordinates in $B$ to Standard Coordinates
Why? The coordinates $(a, b)_B$ say "take $a$ steps along $\vec c_1$ and $b$ steps along $\vec c_2$." Since $\vec c_1$ and $\vec c_2$ already have ordinary components, adding those steps gives the ordinary components of $\vec w$.
An underwater drone moves in an environment with ocean currents. Its autonomous navigation system uses a basis formed by the vectors $\vec{c}1 = (1, 2)$ and…
An underwater drone moves in an environment with ocean currents. Its autonomous navigation system uses a basis formed by the vectors $\vec{c}_1 = (1, 2)$ and $\vec{c}_2 = (-2, 3)$. The drone records a displacement $\vec{w}$ whose coordinates in this basis are $(4, -1)_B$. Calculate the coordinates of the vector $\vec{w}$ in the standard basis.
Find all vectors $\vec{x} = (x_1, x_2)$ which, expressed as a linear combination of the basis $B = \{(1, -1), (2, 1)\}$, have coordinates $(a, b)B$ such that $a = b$…
Find all vectors $\vec{x} = (x_1, x_2)$ which, expressed as a linear combination of the basis $B = \{(1, -1), (2, 1)\}$, have coordinates $(a, b)_B$ such that $a = b$ and, furthermore, satisfy that their modulus in the standard basis is exactly $\sqrt{45}$.
Finding Coordinates in a Basis
Now go the other way. You know $\vec w$ in the standard basis and want $(a, b)_B$. Write the same equation, but with $a$ and $b$ unknown, and solve the system. Elimination works well.
Because $B$ is a basis (independent vectors), the system always has exactly one solution. That is the algebraic meaning of "every vector has unique coordinates." Word problems fit this pattern too: two kinds of trucks with capacity vectors $\vec a$ and $\vec b$ and an order $\vec p$ lead to $\vec p = x\vec a + y\vec b$.
In a video game physics engine, a character's movement is defined in a basis $B = \{\vec{u}, \vec{v}\}$, where $\vec{u} = (2, 1)$ and $\vec{v} = (-1, 3)$. If the…
In a video game physics engine, a character's movement is defined in a basis $B = \{\vec{u}, \vec{v}\}$, where $\vec{u} = (2, 1)$ and $\vec{v} = (-1, 3)$. If the character must move to the point indicated by the vector $\vec{d} = (8, -3)$ with respect to the standard basis, what will be the coordinates of the displacement vector in the basis $B$?
A logistics company uses two types of vans. The load capacity of a type A van is represented by the vector $\vec{a} = (3, 2)$, indicating 3 pallets of fruit and 2 of…
A logistics company uses two types of vans. The load capacity of a type A van is represented by the vector $\vec{a} = (3, 2)$, indicating 3 pallets of fruit and 2 of vegetables. Type B is represented by $\vec{b} = (1, 4)$. If a shopping center places an exact order of 11 pallets of fruit and 14 of vegetables, represented by $\vec{p} = (11, 14)$, find the scalars of the linear combination $\vec{p} = x\vec{a} + y\vec{b}$ to determine how many trips each van needs to make.
Two tugboats pull a broken-down ship with forces $\vec{F}1 = (300, 400)$ N and $\vec{F}2 = (400, -300)$ N. A third tugboat joins in applying a force of…
Two tugboats pull a broken-down ship with forces $\vec{F}_1 = (300, 400)$ N and $\vec{F}_2 = (400, -300)$ N. A third tugboat joins in applying a force of $\vec{F}_3 = (1000, 500)$ N. Express the force $\vec{F}_3$ as a linear combination of the vectors $\vec{F}_1$ and $\vec{F}_2$, finding the coordinates of $\vec{F}_3$ in the basis $B = \{\vec{F}_1, \vec{F}_2\}$.
Back to the start: suppose a diagonal avenue runs along $(1, 1)$ and a cross street along $(1, 0)$. To reach the point $(3, 5)$ you solve $a(1, 1) + b(1, 0) = (3, 5)$. The second component gives $a = 5$, then $5 + b = 3$ gives $b = -2$. So you go 5 blocks along the avenue and 2 blocks back west along the cross street. Same destination, different directions.
Changing Between Two Bases
Sometimes a new basis is described using the vectors of an old one, for example $\vec c_1 = \vec a_1 + 2\vec a_2$. Two questions come up: is the new pair really a basis, and how do coordinates convert?
Is it a basis? Read off the coordinates of the new vectors in the old basis and apply the determinant test to them. Or use the definition: the pair is independent if the only combination equal to $\vec 0$ is the trivial one.
Converting coordinates. Substitute, then collect terms.
To go from $A$-coordinates to $C$-coordinates, you would solve a system, exactly as in the previous section.
Prove analytically that if $B = \{\vec{u}, \vec{v}\}$ is a basis for the vectors of the plane, then the set $B' = \{\vec{u} + \vec{v}, \vec{u} - \vec{v}\}$ also forms a…
Prove analytically that if $B = \{\vec{u}, \vec{v}\}$ is a basis for the vectors of the plane, then the set $B' = \{\vec{u} + \vec{v}, \vec{u} - \vec{v}\}$ also forms a basis. If a vector $\vec{w}$ has coordinates $(3, 5)$ in the basis $B$, find its coordinates in the new basis $B'$.
In the study of a two-dimensional crystal lattice, two bases are defined for the plane: the primitive basis $B_1 = \{\vec{a}1, \vec{a}2\}$ and a secondary basis…
In the study of a two-dimensional crystal lattice, two bases are defined for the plane: the primitive basis $B_1 = \{\vec{a}_1, \vec{a}_2\}$ and a secondary basis $B_2 = \{\vec{b}_1, \vec{b}_2\}$. Knowing that $\vec{b}_1 = 2\vec{a}_1 - \vec{a}_2$ and $\vec{b}_2 = \vec{a}_1 + 3\vec{a}_2$, find the coordinates in the basis $B_1$ of a point defect whose position vector in the basis $B_2$ is $\vec{d} = (-1, 2)_{B_2}$.
Vectors in Geometric Figures
In a figure, you express vectors in terms of two known side vectors by walking along edges, tip to tail. Two facts do most of the work:
Why? "End minus start" is the displacement. For example, from $P(1, 1)$ to $Q(4, 3)$ you move $3$ right and $2$ up, which is $(4 - 1, 3 - 1)$.
For a point question, remember that a point is its own position from the origin: if $\overrightarrow{PR}$ is known, then $R = P + \overrightarrow{PR}$.
Given the parallelogram $ABCD$ in the figure, express the vectors corresponding to the diagonals $\vec{AC}$ and $\vec{BD}$ as a linear combination of the side vectors…
Given the parallelogram $ABCD$ in the figure, express the vectors corresponding to the diagonals $\vec{AC}$ and $\vec{BD}$ as a linear combination of the side vectors $\vec{u} = \vec{AB}$ and $\vec{v} = \vec{AD}$.
In the regular hexagon $ABCDEF$ with center $O$ shown in the figure, the basis vectors $\vec{u} = \vec{AB}$ and $\vec{v} = \vec{AF}$ are defined. Express the vectors…
In the regular hexagon $ABCDEF$ with center $O$ shown in the figure, the basis vectors $\vec{u} = \vec{AB}$ and $\vec{v} = \vec{AF}$ are defined. Express the vectors $\vec{AC}$, $\vec{AD}$, $\vec{AE}$ and $\vec{BC}$ as a linear combination of the basis $B = \{\vec{u}, \vec{v}\}$.
Given the points $A(1, 2)$, $B(4, -1)$ and $C(x, y)$, determine the coordinates of point $C$ knowing that the vectors $\vec{u} = \vec{AB}$ and $\vec{v} = \vec{AC}$ form…
Given the points $A(1, 2)$, $B(4, -1)$ and $C(x, y)$, determine the coordinates of point $C$ knowing that the vectors $\vec{u} = \vec{AB}$ and $\vec{v} = \vec{AC}$ form a basis for the plane in which the vector $\vec{w} = (0, -6)$ has coordinates $(2, -1)_B$.
Magnitude and Angles in a Basis
When a vector is given by coordinates in a basis that is not the standard one, you cannot just use $\sqrt{a^2 + b^2}$. You need what you know about the basis vectors. Two tools:
Why the second one? $a\vec e_1$ and $b\vec e_2$ are perpendicular, so they form the legs of a right triangle and the Pythagorean Theorem applies. The cross term $2ab\,\vec e_1 \cdot \vec e_2$ vanishes because $\vec e_1 \cdot \vec e_2 = 0$.
Check with actual vectors: $\vec e_1 = (3, 0)$, $\vec e_2 = (0, 2)$ give $\vec v = (6, 2)$, with $|\vec v| = \sqrt{40}$ ✓.
Conditions on unknown components work the same way: write one equation for each condition and solve.
Find the values of $a$ and $b$ so that the vectors $\vec{u} = (a, 3)$ and $\vec{v} = (-2, b)$ form an orthogonal basis, also knowing that the modulus of the vector…
Find the values of $a$ and $b$ so that the vectors $\vec{u} = (a, 3)$ and $\vec{v} = (-2, b)$ form an orthogonal basis, also knowing that the modulus of the vector $\vec{u}$ is exactly $5$.
Let $\vec{u}$ and $\vec{v}$ be two non-zero vectors such that the angle they form is $60^\circ$, with $|\vec{u}| = 2$ and $|\vec{v}| = 3$. If the vector…
Let $\vec{u}$ and $\vec{v}$ be two non-zero vectors such that the angle they form is $60^\circ$, with $|\vec{u}| = 2$ and $|\vec{v}| = 3$. If the vector $\vec{w} = m\vec{u} + n\vec{v}$ is defined, calculate the relationship that must exist between the parameters $m$ and $n$ so that the vector $\vec{w}$ is perpendicular to the vector $\vec{u}$.
The arm of an industrial robot moves in a plane defined by a basis $B = \{\vec{e}1, \vec{e}2\}$ which is orthogonal but not orthonormal, where $|\vec{e}1| = 2$ and…
The arm of an industrial robot moves in a plane defined by a basis $B = \{\vec{e}_1, \vec{e}_2\}$ which is orthogonal but not orthonormal, where $|\vec{e}_1| = 2$ and $|\vec{e}_2| = 4$. A trajectory command instructs it to move according to the vector $\vec{v} = 3\vec{e}_1 - \frac{1}{2}\vec{e}_2$. Calculate the exact modulus of the displacement vector $\vec{v}$ and the cosine of the angle it forms with the basis vector $\vec{e}_1$.