Subspaces

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Worksheet on identifying subspaces.

Because spanning is an operation on sets it is natural to ask how it relates to set operations. Given a vector space $V$ and $S, T \subset V$ , select all that are true:

If $S \subset T$ , then $[S]$ is a subspace of $[T]$ . $[ S \cap T ] = [S ] \cap [T]$ $[S \cup T ] = [S] \cup [T]$ $[S^c] = [S]^c$ The set $\left \{ f:\mathbb {R} \to \mathbb {R}: f(x) = f(-x) \right \}$ is a subspace.

Which of the following sets of vectors are linearly independent.

$\left \{\begin {pmatrix} 1 \\ 2 \end {pmatrix} , \begin {pmatrix} 1 \\ 3 \end {pmatrix}\right \}$ $\left \{\begin {pmatrix} 4 \\ 5 \\ -1 \end {pmatrix} , \begin {pmatrix} 8 \\ 10 \\ -2 \end {pmatrix}\right \}$ $\left \{\begin {pmatrix} 0 \\ 0 \end {pmatrix} \right \}$ $\left \{ x+1, (x+1)^2 \right \}$ $\left \{x^2+x+2, x+2, x^2 \right \}$

A Linearly Independent Set that Spans a Vector Space (or a Subspace) is called a basis. Determine which of the following sets of vectors are bases of the indicated vector space.

$\langle \begin {pmatrix} 1 \\ -1 \\0 \end {pmatrix}, \begin {pmatrix} 1 \\ 0 \\ 1 \end {pmatrix} \rangle$ is a basis for $\mathbb {R}^3$ $\langle \begin {pmatrix} 1 \\ -1 \\ 1 \end {pmatrix}, \begin {pmatrix} 1 \\ 1 \\ 1 \end {pmatrix}, \begin {pmatrix} -1 \\ 1 \\ 1 \end {pmatrix}, \begin {pmatrix} 1 \\ 2 \\ 3 \end {pmatrix} \rangle$ is a basis for $\mathbb {R}^3$ $\langle \begin {pmatrix} 1 \\ -1 \\ 1 \end {pmatrix}, \begin {pmatrix} 0 \\ 1 \\ 1 \end {pmatrix}, \begin {pmatrix} 1 \\ 2 \\ 3 \end {pmatrix} \rangle$ a basis for $\mathbb {R}^3$ $\langle x^2 - x + 1, x - 1, x+1 \rangle$ a basis for $P^2$ $\langle x + x^2, x - x^2 \rangle$ a basis for $P^2$