Linear maps

Write the matrix associated with the linear map $f:{\mathbb{R}}^2\to {\mathbb{R}}^3$

$$f(x,y)=(2x-y,x+y,3y)$$

with respect to the standard bases and compute $f(1,-2)$.

Solution available to those who have unlocked the subject.

Determine for which values of $k\in \mathbb{R}$ the following function is linear:

$$f_k(x,y)=(x+k{y}^2,2x-y).$$

Solution available to those who have unlocked the subject.

Find the kernel and the image of the endomorphism of ${\mathbb{R}}^3$

$$f(x,y,z)=(x-2y,y+3z,x+z).$$

Solution available to those who have unlocked the subject.

Prove that $f(x,y,z)=(x+y,y+2z,z)$ is an isomorphism of ${\mathbb{R}}^3$ and find $f^{-1}$.

Solution available to those who have unlocked the subject.

Find a basis of the kernel and the Cartesian equation of the image of

$$f(x,y,z)=(x-y,y-z,x-z).$$

Solution available to those who have unlocked the subject.

Let $f:{\mathbb{R}}^4\to {\mathbb{R}}^2$, $f(x,y,z,w)=(x+y-z,y+z+w)$. Determine whether $f$ is surjective and find a basis of the kernel.

Solution available to those who have unlocked the subject.

Find the linear map $f:{\mathbb{R}}^2\to {\mathbb{R}}^3$ such that

$$f(1,2)=(1,0,1),f(0,1)=(2,1,-1).$$

Solution available to those who have unlocked the subject.

Let $T:{\mathbb{R}}_2[x]\to {\mathbb{R}}_2[x]$, $T(p)=p+p^{\prime }$. Write the matrix of $T$ with respect to the basis $(1,x,x^2)$, determine whether $T$ is an isomorphism and find the polynomial $p$ such that $T(p)=x^2$.

Solution available to those who have unlocked the subject.

Determine whether the matrices

$$A=\left(\begin{array}{cc}1& 2\\ 3& 4\end{array}\right),B=\left(\begin{array}{cc}2& 0\\ 1& 3\end{array}\right)$$

are similar.

Solution available to those who have unlocked the subject.

Let $f:{\mathbb{R}}^2\to {\mathbb{R}}^3$, $f(x,y)=(x,y,x+y)$. Find $M_{\mathcal{C}}^{\mathcal{B}}(f)$ with

$$\mathcal{B}=((1,1),(1,-1)),\mathcal{C}=((1,0,0),(1,1,0),(1,1,1)).$$

Solution available to those who have unlocked the subject.

Let $f(x,y)=(x+y,x-y)$ and $g(x,y)=(2x,x+y)$. Write $g\circ f$, determine whether it is an isomorphism and, if so, find its inverse.

Solution available to those who have unlocked the subject.

Let $f_k$ be the endomorphism of ${\mathbb{R}}^3$ with matrix

$$A_k=\left(\begin{array}{ccc}k& 1& 1\\ 1& k& 1\\ 0& 1& 1\end{array}\right).$$

Find the kernel and the image as $k\in \mathbb{R}$ varies.

Solution available to those who have unlocked the subject.

Determine whether there exists a linear map $f:{\mathbb{R}}^2\to {\mathbb{R}}^3$ such that

$$f(1,0)=(1,1,0),f(1,1)=(1,0,1),f(2,1)=(2,1,1);$$

if so, determine whether it is unique and whether it is injective.

Solution available to those who have unlocked the subject.