Question

Determine if the given set W is a subspace of the given vector space V. If so, find a possible basis for W and give the dimension of W. $\begin{pmatrix} x^2 - y^2 & x + y \ x - y & xy \end{pmatrix}$ : $x, y \in \mathbb{R}$} (a) V = M_{2x2}(\mathbb{R}), W = \left\{ \begin{pmatrix} x^2 - y^2 & x + y \ x - y & xy \end{pmatrix} : x, y \in \mathbb{R} \right\} (b) V = M_{3x3}(\mathbb{R}), W = \{A \in M_{3x3}(\mathbb{R}) : A^T = -A\}, the set of all skew-symmetric 3 \times 3 matrices with real coefficients (c) V = \mathbb{R}_2(X), W = \{(2a - 3b + 1) + (-2a + 5b)X + (2a + b)X^2 : a, b \in \mathbb{R}\} (d) V = \mathbb{R}_4(X), W = \{a_0 + a_1X + a_2X^2 + a_3X^3 + a_4X^4 : a_0, a_1, ..., a_4 \in \mathbb{R}, a_0 + a_1 + a_2 + a_3 + a_4 = 0\}

          Determine if the given set W is a subspace of the given vector space V. If so, find a possible
basis for W and give the dimension of W.
$\begin{pmatrix} x^2 - y^2 & x + y \ x - y & xy \end{pmatrix}$ : $x, y \in \mathbb{R}$}
(a) V = M_{2x2}(\mathbb{R}), W = \left\{ \begin{pmatrix} x^2 - y^2 & x + y \ x - y & xy \end{pmatrix} : x, y \in \mathbb{R} \right\}
(b) V = M_{3x3}(\mathbb{R}), W = \{A \in M_{3x3}(\mathbb{R}) : A^T = -A\}, the set of all skew-symmetric 3 \times 3
matrices with real coefficients
(c) V = \mathbb{R}_2(X), W = \{(2a - 3b + 1) + (-2a + 5b)X + (2a + b)X^2 : a, b \in \mathbb{R}\}
(d) V = \mathbb{R}_4(X),
W = \{a_0 + a_1X + a_2X^2 + a_3X^3 + a_4X^4 : a_0, a_1, ..., a_4 \in \mathbb{R}, a_0 + a_1 + a_2 + a_3 + a_4 = 0\}

$Determine if the given set W is a subspace of the given vector space V. If so, find a possible basis for W and give the dimension of W. < p m a t r i x > : x, y ∈ℝ (a) V = M2x2(ℝ), W = { < p m a t r i x > : x, y ∈ℝ } (b) V = M3x3(ℝ), W = {A ∈M3x3(ℝ) : A^T = -A}, the set of all skew-symmetric 3 ×3 matrices with real coefficients (c) V = ℝ2(X), W = {(2a - 3b + 1) + (-2a + 5b)X + (2a + b)X^2 : a, b ∈ℝ} (d) V = ℝ4(X), W = {a0 + a1X + a2X^2 + a3X^3 + a4X^4 : a0, a1, ..., a4 ∈ℝ, a0 + a1 + a2 + a3 + a4 = 0}$

Added by Leslie M.

Question

Please give Ace some feedback