Find $3\times 3$ matrices representing the transformations of $\mathbb R^3$ which:a) project every vector onto $x\text{-}y$ plane;b) reflect every vector through $x\text{-}y$ plane;c) rotate the $x\text{-}y$ plane through $30^\circ$, leaving…
For each linear transformation below find its matrix a) $T:\mathbb R^2\to \mathbb R^3$ defined by $$T(x,y)^T=(x+2y,2x-5y,7y)^T;$$ Solution: We have $$T\begin{pmatrix}x\\ y\end{pmatrix}=x\begin{pmatrix} 1\\ 2\\ 0\end{pmatrix}+y\begin{pmatrix} 2\\ -5\\ 7\end{pmatrix},$$ hence the matrix…
Let a linear transformation in $\mathbb R^2$ be the reflection in the line $x_1=x_2$. Find its matrix. Solution: Let us think $x_1=x$ and $x_2=y$. In the $(x,y)$-coordinate plane, the graph…
Multiply a) $\begin{pmatrix} 1 & 2 & 3\\ 4 &5 & 6\end{pmatrix}\begin{pmatrix} 1 \\ 3\\ 2 \end{pmatrix}$; b) $\begin{pmatrix} 1 & 2 \\ 0 & 1\\ 2 & 0\end{pmatrix}\begin{pmatrix} 1…
Is it possible that vectors ${\bf v}_1$, ${\bf v}_2$, ${\bf v}_3$ are linearly dependent, but the vectors ${\bf w}_1={\bf v}_1+{\bf v}_2$, ${\bf w}_2={\bf v}_2+{\bf v}_3$, ${\bf w}_3={\bf v}_3+{\bf v}_1$ are…
Let a system of vectors ${\bf v}_1$, ${\bf v}_2$, $\cdots$, ${\bf v}_r$ be linearly independent but not generating. Show that it is possible to find a vector ${\bf v}_{r+1}$ such that the system ${\bf v}_1$, ${\bf v}_2$, $\cdots$, ${\bf v}_r$, ${\bf v}_{r+1}$ is linearly independent. Hint: Take for ${\bf v}_{r+1}$ any vector that cannot be represented as a linear combination $\sum_{k=1}\alpha_k{\bf v}_{k}$ and show that the system ${\bf v}_1$, ${\bf v}_2$, $\cdots$, ${\bf v}_r$, ${\bf v}_{r+1}$ is linearly independent. (more…)
Write down a basis for the space of
a) $3\times 3$ symmetric matrices;
b) $n\times n$ symmetric matrices;
c) $n\times n$ anti-symmetric ($A^T = -A$) matrices; (more…)
Recall, that a matrix is called symmetric if $A^T = A$. Write down a basis in the space of symmetric $2\times 2$ matrices (there are many possible answers). How many elements are in the basis?
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True or false:
a) Any set containing a zero vector is linearly dependent;
b) A basis must contain ${\bf 0}$;
c) subsets of linearly dependent sets are linearly dependent;
d) subsets of linearly independent sets are linearly independent;
e) If $\alpha_1{\bf v}_1+\alpha_2{\bf v}_2+\cdots+\alpha_n{\bf v}_n=0$ then all scalars $\alpha_k$ are zero. (more…)
Find a basis in the space of $3\times 2$ matrices $M_{3\times 2}$. (more…)