Linear Algebra: Span and Linear Independence

Using the criteria for linear dependence without division, determine if the columns of a given 2x2 matrix are linearly dependent.

Determine if the set of vectors $(1, -2, 0)$, $(4, 0, 8)$, $(3, -1, 5)$ is linearly independent or dependent by performing row reduction.

Are the vectors $\mathbf{v}_1 = (1, 2, 3)$, $\mathbf{v}_2 = (2, -1, 4)$, and $\mathbf{v}_3 = (0, 5, 2)$ linearly independent?

Given a homogeneous system of linear equations in the form $A \mathbf{x} = 0$, describe the solution set in parametric vector form.

Put these three vectors into a matrix, row reduce it, and identify how many pivots we get to determine the dimension of the span.

Find the span of these two vectors in $\mathbb{R}^4$ using the row reduction technique to determine the dimension.

Determine the dimension of the span for these three vectors in $\mathbb{R}^3$ by putting them in a matrix and row reducing to find the number of pivots.

Given three vectors from $\mathbb{R}^3$, which are (2, 1, -1), (0, 2, 2), and (-1, -1, -1), determine their span by forming the linear combination $a\mathbf{v_1} + b\mathbf{v_2} + c\mathbf{v_3}$ where $a, b,$ and $c$ are scalars.

Is a given vector $W = \begin{bmatrix} 1 \\ 2 \\ 4 \end{bmatrix}$ in the span of two vectors $U = \begin{bmatrix} 1 \\ 0 \\ 1 \end{bmatrix}$ and $V = \begin{bmatrix} 0 \\ 1 \\ 1 \end{bmatrix}$?