# Tagged: linear dependent

## Problem 279

Determine conditions on the scalars $a, b$ so that the following set $S$ of vectors is linearly dependent.
\begin{align*}
S=\{\mathbf{v}_1, \mathbf{v}_2, \mathbf{v}_3\},
\end{align*}
where
$\mathbf{v}_1=\begin{bmatrix} 1 \\ 3 \\ 1 \end{bmatrix}, \mathbf{v}_2=\begin{bmatrix} 1 \\ a \\ 4 \end{bmatrix}, \mathbf{v}_3=\begin{bmatrix} 0 \\ 2 \\ b \end{bmatrix}.$

## Problem 137

Let $P_n(\R)$ be the vector space over $\R$ consisting of all degree $n$ or less real coefficient polynomials. Let
$U=\{ p(x) \in P_n(\R) \mid p(1)=0\}$ be a subspace of $P_n(\R)$.

Find a basis for $U$ and determine the dimension of $U$.

## Problem 120

Suppose that $\mathbf{v}_1, \mathbf{v}_2, \dots, \mathbf{v}_r$ are linearly dependent $n$-dimensional real vectors.

For any vector $\mathbf{v}_{r+1} \in \R^n$, determine whether the vectors $\mathbf{v}_1, \mathbf{v}_2, \dots, \mathbf{v}_r, \mathbf{v}_{r+1}$ are linearly independent or linearly dependent.

## Problem 15

Let $p_1(x), p_2(x), p_3(x), p_4(x)$ be (real) polynomials of degree at most $3$. Which (if any) of the following two conditions is sufficient for the conclusion that these polynomials are linearly dependent?

(a) At $1$ each of the polynomials has the value $0$. Namely $p_i(1)=0$ for $i=1,2,3,4$.

(b) At $0$ each of the polynomials has the value $1$. Namely $p_i(0)=1$ for $i=1,2,3,4$.

(University of California, Berkeley)