Linear Algebra: Subspaces Basis and Dimension

Check if a set of vectors in $\mathbb{R}^3$ consisting of (1, 0, 0), (0, 1, 0), and (0, 0, 1) form a basis for $\mathbb{R}^3$.

Determine if the given set of four matrices, with specific ones and zeros, span $R^{2x2}$ and form a basis.

$\begin{bmatrix}1 & 1 \\0 & 0\end{bmatrix}$ $\begin{bmatrix}0 & 0 \\1 & 1\end{bmatrix}$ $\begin{bmatrix}1 & 0 \\0 & 1\end{bmatrix}$ $\begin{bmatrix}0 & 1 \\1 & 1\end{bmatrix}$

Check comprehension related to vector spaces, subspaces, span, linear independence, basis, and dimension.

Determine the column space of the given matrix $A$.

Consider a matrix $A$ and determine the column space, which is the set of all vectors $\mathbf{b}$ such that there exists a vector $\mathbf{x}$ where $A\mathbf{x} = \mathbf{b}$.

Given a set of four vectors in $\mathbb{R}^2$, put them as columns of a matrix, row reduce, and identify the number of pivots to determine the dimension of the span.

Let V be a vector space. Verify whether a subset S, which is made of vectors of the form (x, 0, -x), is a subspace of V by checking the properties of closure.

Given the set U consisting of vectors from $\mathbb{R}^3$ defined by the equations:

$a = 2r - s, \quad b = 3r, \quad c = r + s$ where $r, s$ are real numbers, determine whether U is a subspace of $\mathbb{R}^3$.

Let $W = \{ [a + 2b, a - b, 3b] \mid a, b \in \mathbb{R} \}$. Determine if $W$ is a subspace of $\mathbb{R}^3$.