*Read solution In this problem, we use the following vectors in $\R^2$.\[\mathbf=\begin 1 \ 0 \end, \mathbf=\begin 1 \ 1 \end, \mathbf=\begin 2 \ 3 \end, \mathbf=\begin 3 \ 2 \end, \mathbf=\begin 0 \ 0 \end, \mathbf=\begin 5 \ 6 \end.\] For each set $S$, determine whether $\Span(S)=\R^2$.Read solution Calculate the following expressions, using the following matrices: \[A = \begin 2 & 3 \ -5 & 1 \end, \qquad B = \begin 0 & -1 \ 1 & -1 \end, \qquad \mathbf = \begin 2 \ -4 \end\] (a) $A B^\trans \mathbf \mathbf^\trans$. Read solution Let \[\mathbf_1=\begin 1 \ 2 \ 0 \end, \mathbf_2=\begin 1 \ a \ 5 \end, \mathbf_3=\begin 0 \ 4 \ b \end\] be vectors in $\R^3$.*

(a) Show that if $\mathbf_1, \mathbf_2$ are linearly dependent vectors, then the vectors $A\mathbf_1, A\mathbf_2$ are also linearly dependent.

(b) If $\mathbf_1, \mathbf_2$ are linearly independent vectors, can we conclude that the vectors $A\mathbf_1, A\mathbf_2$ are also linearly independent?

Set the $n\times n$ matrix $B=[A_1, A_2, \dots, A_, A\mathbf]$, where $A_i$ is the $i$-th column vector of $A$.

Prove that $B$ is a singular matrix for any choice of $\mathbf$. Assume that every vector $\mathbf$ in $\R^n$ is an eigenvector for some eigenvalue of $A$.

Since the plane is moving South with respect to the air, the new components along ground are going to be $192-23\cos$mph towards South and $23\sin$mph towards East.

Let $A=\begin 1 & 0 & 3 & -2 \ 0 &3 & 1 & 1 \ 1 & 3 & 4 & -1 \end$.

Read solution For each of the following matrix $A$, prove that $\mathbf^A\mathbf \geq 0$ for all vectors $\mathbf$ in $\R^2$. Prove that there exists $\lambda\in \R$ such that $A=\lambda I$, where $I$ is the $n\times n$ identity matrix.

Also, determine those vectors $\mathbf\in \R^2$ such that $\mathbf^A\mathbf=0$. Read solution Problem 1 Let $W$ be the subset of the $-dimensional vector space $\R^3$ defined by \[W=\left\.\] (a) Which of the following vectors are in the subset $W$? \[(1) \begin 0 \ 0 \ 0 \end \qquad(2) \begin 1 \ 2 \ 2 \end \qquad(3)\begin 3 \ 0 \ 0 \end \qquad(4) \begin 0 \ 0 \end \qquad(5) \begin 1 & 2 & 4 \ 1 &2 &4 \end \qquad(6) \begin 1 \ -1 \ -2 \end.\] (b) Determine whether $W$ is a subspace of $\R^3$ or not.

If $\Span(S)\neq \R^2$, then give algebraic description for $\Span(S)$ and explain the geometric shape of $\Span(S)$.

(a) $S=$ (b) $S=$ (c) $S=$ (d) $S=$ (e) $S=$ (f) $S=$ (g) $S=$ Read solution Let $\mathbf$ and $\mathbf$ be two $n \times 1$ column vectors.

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