Vector spaces, bases and dimension

Determine whether

$$W=\{(x,y,z)\in {\mathbb{R}}^3:x+y=z,x=3y\}$$

is a subspace of ${\mathbb{R}}^3$ and, if so, find a basis of it.

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Determine whether

$$W=\{(x,y)\in {\mathbb{R}}^2:x^2=y^2\}$$

is a subspace of ${\mathbb{R}}^2$.

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Verify that

$$W=\{p\in {\mathbb{R}}_3[x]:p(-x)=p(x)\text{for every }x\}$$

is a subspace of ${\mathbb{R}}_3[x]$ and find a basis and the dimension of it.

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Determine whether the vectors $v_1=(1,2,3)$, $v_2=(2,1,0)$, $v_3=(1,-1,-3)$ are linearly independent; if not, write one of them as a linear combination of the others.

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Determine for which values of $k\in \mathbb{R}$ the vectors

$$v_1=(1,k,0),v_2=(0,1,k),v_3=(k,0,1)$$

are linearly independent.

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Determine whether the matrices

$$A_1=\left(\begin{array}{cc}1& 1\\ 0& 0\end{array}\right),A_2=\left(\begin{array}{cc}0& 1\\ 1& 0\end{array}\right),A_3=\left(\begin{array}{cc}1& 0\\ 0& 1\end{array}\right),A_4=\left(\begin{array}{cc}0& 0\\ 1& 1\end{array}\right)$$

form a basis of $M_2(\mathbb{R})$, and find the dimension of the subspace they span.

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Find a basis and the dimension of the subspace

$$W=\{(x_1,x_2,x_3,x_4)\in {\mathbb{R}}^4:x_1-x_2+2x_3-x_4=0\}.$$

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Verify that $\mathcal{B}=(1,1+x,1+x+x^2)$ is a basis of ${\mathbb{R}}_2[x]$ and find the coordinates of $p(x)=3-2x+5x^2$ with respect to $\mathcal{B}$.

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Let $W=⟨(1,1,2),(2,0,1),(0,2,3),(3,1,3)⟩\subseteq {\mathbb{R}}^3$. Extract a basis of $W$ from the given generators, and find $\dim W$ and a Cartesian equation of $W$.

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Complete the vectors $v_1=(0,0,1,1)$ and $v_2=(1,2,0,1)$ to a basis of ${\mathbb{R}}^4$ by adding vectors of the standard basis.

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Find Cartesian equations of the subspace

$$W=⟨(1,1,0,0),(0,1,1,1)⟩\subseteq {\mathbb{R}}^4.$$

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In ${\mathbb{R}}^4$ consider

$$U:x_1+x_2+x_3+x_4=0,W=⟨(1,-1,0,0),(1,1,1,1),(0,0,1,-1)⟩.$$

Find a basis of $U\cap W$ and the dimension of $U+W$.

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In ${\mathbb{R}}^3$ consider the plane $U:x+y-z=0$ and the line $W_k=⟨(1,k,1)⟩$, with $k\in \mathbb{R}$. Determine for which values of $k$ we have

$${\mathbb{R}}^3=U\oplus W_k.$$

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