Solutions to the practice problems

Look here for some solutions to the practice problems

The third midterm

• Solutions to the practice problems
• The third midterm
• Things to know
• Practice problems
  • Computations with matrices
  • Compute the matrix of $T:V\to V$ with respect to a given basis of $V$
  • Compute some determinants
  • True/False?

Location

In class, B102 during regular lecture time Thursday November 24

Topics

Linear Transformations

Matrices of linear Transformations

Determinants

Format

The exam will have the following three kind of problems:

• Short answer True/False questions
• Three computational problems, using the concepts covered above.

Things to know

• How to compute the matrix of a linear transformation $V\stackrel{T}{\longrightarrow}W$ with respect to given ordered bases of $V$ and $W$.
• How to multiply two matrices
• Know an example of a matrix $A$ with $A\neq O$ but $A^2=O$
• How to find the inverse of a matrix using row reduction/Gaussian elimination
• Find the sign of a permutation
• Know the important determinant properties: there is a good summary in §4.4, pages 232—236 of the text book.
• How to compute a determinant
  • by adding rows or columns repeatedly
  • by expanding along a row or column
  • by using the definition (not advisable except for $2\times 2$ determinants)
• Uses of the determinant:
  • The cofactor matrix, finding the inverse of a matrix, and Cramer's rule
  • Is a matrix invertible?
  • Is a set of vectors a basis for $\F^n$?

Practice problems

Computations with matrices

a. Matrix products: Homework 5, problem 4 & problem 5

b. Consider the following matrices:

$A= \mat 1 & 3 & -1 \\ 2 & 0 & 1\rix, \qquad B= \mat 3 & 3\rix$

Compute each of the following matrix expressions, if they are defined:

1. $AB$, $BA$, $AA^\top$, $A^\top A$
2. $BB^\top$, $B^\top B$
3. $A^2$
4. $(AA^\top)^2$
5. $A^\top B^\top B A$
6. $AA^\top+BB^\top$
7. $AA^\top+B^\top B$

Compute the matrix of $T:V\to V$ with respect to a given basis of $V$

a. $V=\cP_4(\R)$, the standard basis $\{1,x,x^2,x^3,x^4\}$, $Tf(x) = f''(x)-xf'(x)$

b. $V=\R^2$ and $T:\R^2\to\R^2$ is given by $T(x_1, x_2) = (2x_1, x_2)$.

• Find the matrix of $T$ with respect to the standard basis $\{e_1, e_2\}$ of $\R^2$
• Find the matrix of $T$ with respect to the basis $\{u,v\}$ where $u=\binom{1}{2}$, $v=\binom{2}{1}$,

c. $V$ is the vector space

$V=\{(x_1,x_2,x_3)\in\R^3 \mid x_1+x_2+x_3=0\},$

with basis $\{u,v\}$ where

$u = \tmat 1\\-1\\0\trix, \qquad v=\tmat 0\\1\\-1 \trix,$

and $T:V\to V$ is the linear transformation

$T(x_1, x_2, x_3) = (x_3,x_1,x_2).$

Compute some determinants

a. From the book, page 237,Problem 4

b. For which value of $a\in\R$ is $\tmat 1 &a &1\\a & 1 & 1 \\ 1 & 1 & 1 \trix$ invertible?

c. Compute the determinant of $\tmat 1& 1 & 1& 1& 1\\ 1& 2 & 3& 4& 5\\ 1& 3 & 5& 7& 9\\ 1& 4 & 7& 10& 13\\ 1& 5 & 9& 13& 17 \trix$

d. Let $A=\tmat 1& 2 & 1\\ 2& 0 &-1 \\ -5& 0 &3 \trix$, and let $B=A^{-1}$. Compute the cofactor matrix of $A$. Then use the cofactor formula to compute $b_{11}$, $b_{12}$, and $b_{13}$.

e. Let $A=\tmat 1& 2 & 1\\ 2& 0 &-1 \\ -5& 0 &3 \trix$. Use row reduction (or “Gaussian elimination”) to compute $A^{-1}$.

f. For which values of $t\in\R$ are the vectors

$u=\mat 1 \\ 2 \\ t\rix,\quad v=\mat 2 \\ 1 \\ t\rix,\quad w=\mat 1 \\ t \\ 0\rix$

a basis of $\R^3$?

True/False?

1. For any $k\times \ell$ matrix $A$, the matrix product $AA^\top$ is always defined, no matter what $k$ and $\ell$ are.
2. If $A= \tmat1 & 3 & -1 \\ 2 & 0 & 1\trix$, then $\det(AA^\top) = \det (A) \det (A^\top)$
3. If $A$ is a square matrix then $\det(A^\top A) \geq 0$
4. If $A$ is a square matrix with $A=-A^\top$, then $\det A^\top = \det A$
5. If $A$ is a $3\times 3$ square matrix with $A^\top=-A$ then $\det A=0$
6. If $A$ is a square matrix with $A^2=O$ then $A=O$
7. If $A$ is an $n\times n$ matrix and if there is a nonzero vector $x\in\F^n$ with $Ax=0$ then $\det A = 0$.
8. If $A$ is a square matrix with $A^2=O$ then $\det A=0$.
9. From the book: §4.4, page 236, problem 1—if a T/F question claims some equation, and you think it is wrong, say how the equation can be fixed.