University Linear Algebra — Change of Basis and Matrix Representations Worksheet

All 9 questions

Each question targets one named concept from the sheet. Read them here, or print the PDF — it has working space under each one.

1. Q1The Change of Basis Matrix

   In $P_2$, the space of polynomials of degree at most $2$, let $𝒮=\{1,x,x^2\}$ be the standard basis and $ℬ=\{1,1+x,(1+x{)}^2\}$. The change-of-basis matrix $P_{𝒮\leftarrow ℬ}$ is the matrix whose $j$th column is the $𝒮$-coordinate vector of the $j$th vector of $ℬ$.

   1. Find $P_{𝒮\leftarrow ℬ}$ and $P_{ℬ\leftarrow 𝒮}$.
   2. Use a change-of-basis matrix to find $[p{]}_{ℬ}$ for $p(x)=3-2x+5x^2$.
   3. Find the polynomial $q$ with $[q{]}_{ℬ}=(2,-1,3)$.
2. Q2The Change of Basis Matrix

   Let $W=span\{{\overrightarrow{c}}_1,{\overrightarrow{c}}_2\}$, a plane through the origin in ${ℝ}^4$, where $${\overrightarrow{c}}_1=(1,0,1,2),{\overrightarrow{c}}_2=(0,1,-1,1),{\overrightarrow{b}}_1=(2,1,1,5),{\overrightarrow{b}}_2=(1,-1,2,1).$$ Both $𝒞=\{{\overrightarrow{c}}_1,{\overrightarrow{c}}_2\}$ and $ℬ=\{{\overrightarrow{b}}_1,{\overrightarrow{b}}_2\}$ are bases of $W$. The matrix $P_{𝒞\leftarrow ℬ}$ has $[{\overrightarrow{b}}_j{]}_{𝒞}$ as its $j$th column.

   1. Find $P_{𝒞\leftarrow ℬ}$ and $P_{ℬ\leftarrow 𝒞}$.
   2. A vector $\overrightarrow{x}\in W$ has $[\overrightarrow{x}{]}_{ℬ}=(1,-2)$. Find $[\overrightarrow{x}{]}_{𝒞}$ and $\overrightarrow{x}$ itself, and check your answer.
   3. Find $[\overrightarrow{y}{]}_{ℬ}$ for $\overrightarrow{y}=(3,-3,6,3)$.
3. Q3The Matrix of a Transformation Relative to Given Bases

   Let $T:{ℝ}^3\to {ℝ}^2$ be the linear map $T(x_1,x_2,x_3)=(x_1-x_2+2x_3,3x_1+x_3)$, and take the bases $$ℬ=\{(1,1,0),(0,1,1),(1,0,1)\}\text{of }{ℝ}^3,𝒞=\{(1,1),(1,2)\}\text{of }{ℝ}^2.$$ The matrix $[T{]}_{𝒞\leftarrow ℬ}$ has $[T({\overrightarrow{b}}_j){]}_{𝒞}$ as its $j$th column.

   1. Find $[T{]}_{𝒞\leftarrow ℬ}$.
   2. The vector $\overrightarrow{v}$ has $[\overrightarrow{v}{]}_{ℬ}=(1,2,-1)$. Use your matrix to find $[T(\overrightarrow{v}){]}_{𝒞}$, then $T(\overrightarrow{v})$, and check by computing $\overrightarrow{v}$ and applying $T$ directly.
4. Q4The Matrix of a Transformation Relative to Given Bases

   Let $V$ have basis $ℬ=\{{\overrightarrow{b}}_1,{\overrightarrow{b}}_2,{\overrightarrow{b}}_3\}$ and $W$ have basis $𝒞=\{{\overrightarrow{c}}_1,{\overrightarrow{c}}_2\}$. A linear map $T:V\to W$ has $$[T{]}_{𝒞\leftarrow ℬ}=[\begin{array}{ccc}1& 0& -2\\ 3& 1& 4\end{array}],$$ where the $j$th column is $[T({\overrightarrow{b}}_j){]}_{𝒞}$.

   1. Write $T({\overrightarrow{b}}_2)$ and $T({\overrightarrow{b}}_1-{\overrightarrow{b}}_3)$ in terms of ${\overrightarrow{c}}_1$ and ${\overrightarrow{c}}_2$.
   2. Find $T(2{\overrightarrow{b}}_1-{\overrightarrow{b}}_2+{\overrightarrow{b}}_3)$.
   3. Find a nonzero vector of $\ker T$, written in terms of ${\overrightarrow{b}}_1,{\overrightarrow{b}}_2,{\overrightarrow{b}}_3$.
5. Q5Matrix Representations on Polynomial and Matrix Spaces

   Let $T:P_2\to P_3$ be the linear map $T(p)(x)=(x-2)p(x)$.

   1. Find the matrix of $T$ relative to the standard bases $\{1,x,x^2\}$ of $P_2$ and $\{1,x,x^2,x^3\}$ of $P_3$ (the $j$th column is the coordinate vector of the image of the $j$th basis polynomial).
   2. Find the matrix of $T$ relative to the bases $\{1,x-2,(x-2{)}^2\}$ of $P_2$ and $\{1,x-2,(x-2{)}^2,(x-2{)}^3\}$ of $P_3$.
6. Q6The Change of Basis Formula for a Linear Operator

   The linear operator $T:{ℝ}^3\to {ℝ}^3$ has standard matrix $$A=\left[\begin{array}{ccc}1& 2& 0\\ 0& 1& -1\\ 1& 0& 3\end{array}\right].$$ Let $ℬ=\{(1,0,0),(1,1,0),(1,1,1)\}$ and let $P$ be the matrix with these vectors as columns.

   1. Find $P^{-1}$ and use $[T{]}_{ℬ}=P^{-1}AP$ to find $[T{]}_{ℬ}$.
   2. Check the second column of your answer directly: compute $T(1,1,0)$ and confirm that it equals the combination of the vectors of $ℬ$ that the column prescribes.
7. Q7The Change of Basis Formula for a Linear Operator

   Let $ℬ=\{{\overrightarrow{b}}_1,{\overrightarrow{b}}_2\}$ with ${\overrightarrow{b}}_1=(1,1)$ and ${\overrightarrow{b}}_2=(1,2)$. A linear operator $T$ on ${ℝ}^2$ satisfies $T({\overrightarrow{b}}_1)=2{\overrightarrow{b}}_1-{\overrightarrow{b}}_2$ and $T({\overrightarrow{b}}_2)={\overrightarrow{b}}_1$.

   1. Write down $[T{]}_{ℬ}$, whose columns are $[T({\overrightarrow{b}}_1){]}_{ℬ}$ and $[T({\overrightarrow{b}}_2){]}_{ℬ}$.
   2. Use the change-of-basis formula to find the standard matrix $A$ of $T$.
   3. Find $T(3,5)$ with $A$, and again by writing $(3,5)$ in terms of $ℬ$.
8. Q8Similar Matrices and Similarity Invariants

   Two $n\times n$ matrices $A$ and $B$ are similar if $B=P^{-1}AP$ for some invertible $P$; similar matrices have the same trace, the same determinant and the same rank. For each pair, either name an invariant that shows $A$ and $B$ are not similar, or verify that $B=P^{-1}AP$ for the $P$ given (check $AP=PB$).

   1. $A=\left[\begin{array}{cc}3& 1\\ 2& 4\end{array}\right]$, $B=\left[\begin{array}{cc}5& 2\\ 1& 2\end{array}\right]$
   2. $A=\left[\begin{array}{ccc}1& 0& 2\\ 0& 1& 0\\ 0& 0& 0\end{array}\right]$, $B=\left[\begin{array}{ccc}1& 1& 1\\ 1& 1& 1\\ 0& 0& 0\end{array}\right]$
   3. $A=\left[\begin{array}{cc}3& 1\\ 2& 4\end{array}\right]$, $B=\left[\begin{array}{cc}1& -2\\ 2& 6\end{array}\right]$, $P=\left[\begin{array}{cc}1& 1\\ 0& 1\end{array}\right]$
9. Q9Synthesis — drawing on several topics in this unit

   Let $T:P_2\to P_2$ be the linear operator $T(p)(x)=x{p}^{\prime }(x)$, let $ℰ=\{1,x,x^2\}$ and let $𝒞=\{1,1+x,1+x+x^2\}$. For an operator, $[T{]}_{ℬ}$ has $[T(b_j){]}_{ℬ}$ as its $j$th column.

   1. Find $[T{]}_{ℰ}$.
   2. Find $P=P_{ℰ\leftarrow 𝒞}$ (columns $[c_j{]}_{ℰ}$) and $P^{-1}$, and use $[T{]}_{𝒞}=P^{-1}[T{]}_{ℰ}P$ to find $[T{]}_{𝒞}$.
   3. Check the third column of $[T{]}_{𝒞}$ by computing $T(1+x+x^2)$ directly.
   4. Name two numbers that $[T{]}_{ℰ}$ and $[T{]}_{𝒞}$ must share, and confirm that they do.

The 6 challenge problems for this topic are a separate, paid sheet and are not reproduced here.

The rest of this unit

The worksheet is the practice, and it is free. Two more printable documents cover the same unit and come with the University Linear Algebra bundle: read the notes first, work this sheet, then sit the test closed-book. See what each one covers.

What the Change of Basis and Matrix Representations notes cover  About the Change of Basis and Matrix Representations unit test

How to do every concept on this sheet

This is the part worksheet sites usually leave out. Below is the actual reasoning behind each group of questions — not a full solution set, the decisions that get you to one. Read it before you start, or after you get stuck.

The change-of-basis matrix

Given two bases of the same space, the change-of-basis matrix from B to S converts B-coordinates into S-coordinates. Q1 states the rule for building it: its columns are the S-coordinates of the vectors of B.

In Q1 the target basis is the standard basis of the quadratics, so each column is just the coefficients of one basis polynomial — expand the square first. The matrix in the other direction is the inverse. Part (b) converts standard coordinates into B-coordinates, so use the matrix whose arrow points toward B; part (c) runs the other way.

Q2 has two bases of a plane inside four-space, and neither is standard. Column j of the matrix from B to C is the C-coordinates of the j-th vector of B, so you solve a small system for each b_j — both at once, with the two vectors augmented side by side.

The matrix of a transformation relative to given bases

A linear map from V to W has one matrix for every choice of basis B of V and C of W. Q3 gives the rule: column j is the C-coordinates of T applied to the j-th vector of B.

Q3(b) is what the matrix is for: multiplying B-coordinates of v by it gives C-coordinates of T(v). The question then asks for the check that makes the whole construction trustworthy — compute v itself, apply T directly, and compare.

Q4 gives only the matrix, with abstract bases, and asks you to read it. Column two is the image of the second basis vector; a combination of basis vectors maps to the same combination of columns. Part (c) asks for a non-zero kernel vector: find a non-zero solution of the matrix times a coordinate vector equal to zero, then translate the coordinates back into a combination of b₁, b₂, b₃.

Matrix representations on polynomial spaces

Q5 is a map from quadratics to cubics that multiplies by a fixed linear factor. Part (a) uses the standard bases: multiply each of 1, x and x² by the factor, expand, and record the coefficients as columns. Part (b) uses bases built from powers of the same factor.

The change-of-basis formula for an operator

An operator maps a space to itself, so one basis serves for both sides. If A is its standard matrix and P has the vectors of B as columns, the matrix relative to B is:

Read it from right to left: P converts B-coordinates to standard ones, A applies the map, and P⁻¹ converts back. Q6 applies the formula, then asks you to confirm one column by the definition — compute the image of the second basis vector and check that it is the combination of B that the column says.

Q7 runs the formula backwards. The map is described by what it does to the basis vectors, so its matrix relative to B can be written down with no computation; solving the formula for A gives the standard matrix.

Similar matrices

Two square matrices are similar when one is P⁻¹ times the other times P for some invertible P — that is, when they are matrices of the same operator in two bases. Q8 lists three numbers similar matrices always share, and asks you to use them.

The synthesis question

Q9 follows one operator on quadratics — x times the derivative — through the set. Part (a) is Q5(a): apply the operator to 1, x and x² and record coefficients. Part (b) is Q1 and Q6 together: build P from the second basis in standard coordinates, invert it, and apply the formula. Part (c) is Q6(b)'s check, on the third column. Part (d) returns to Q8: the two matrices are similar by construction, so name two of the invariants Q8 lists and compute them for both.

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Getting the most out of it

Label every matrix with its arrow

Write P_C←B or [T]_C←B, never just P. When you multiply, the inner labels must match — coordinates in B go into a matrix whose right-hand label is B. This one habit prevents most errors in the set.

Columns are coordinate vectors

Every matrix in this set is built the same way: apply something to a basis vector, write the result in coordinates, make it a column. When you are unsure how to start, go back to that sentence.

Check one column by the definition

After using a formula, pick one basis vector, apply the map directly, and confirm the matching column. It takes a minute and catches a wrong inverse or a transposed product.

Avoid inverses when you can

To check B = P⁻¹AP, check AP = PB. To convert into the standard basis, multiply by the matrix of basis vectors directly. Invert only when the question needs the other direction.

Common questions