Show that the set W of all symmetric $(3 \times 3)$ matrices is a subspace of the vector space of all $(3 \times 3)$ matrices. Find a spanning set for W.

Let $\mathcal{V}$ be the subspace of $\mathcal{M}_{22}$ consisting of all symmetric $2 \times 2$ matrices. Let S be the set of nonsingular matrices in $\mathcal{V}$. Find a subset of S that is a basis for $\operatorname{span}(S)$.

A $1.00 \cdot 10^{-6} \ g$ sample of nobelium, $_{102}^{254} \mathrm{No}$, has a half-life of 55 seconds after it is formed. What is the percentage of $_{102}^{254} \mathrm{No}$ remaining at the following times?

(a) 5.0 min after it forms

(b) 1.0 h after it forms

A matrix M is called skew-symmetric if

$$M^{\tau} = -M.$$

Clearly, a skew-symmetric matrix is square. Let F be a field. Prove that the set

$$W_1$$

of all skew-symmetric n x n matrices with entries from F is a subspace of

$$M_{n\times n}(F).$$

Now assume that F is not of characteristic 2 , and let

$$W_2$$

be the subspace of

$$M_{n \times n}(F)$$

consisting of all symmetric n x n matrices. Prove that

$$M_{n \times n}(F)= W_1\oplus W_2.$$

Let $R$ be the set of all matrices of the form

$$\left[\begin{array}{rr}a & -b \\ b & a\end{array}\right]$$

, where $a$ and $b$ are integers. Assume that $R$ is a ring with respect to matrix addition and multiplication. Determine whether $R$ is commutative and identify the unity if $R$ has one.

Let S be the set of all symmetric 2 x 2 matrices with real entries. Show that S is a subspace of $\mathbb{R}^{2 \times 2}$.

Show that the set of all matrices of the form

$$B=\left(\begin{array}{cc} \pm 1 & n \\ 0 & 1 \end{array}\right)$$

, where $n \in \mathbb{Z}_{n}$, is a group isomorphic to $D_{n}$.

Let S be the subspace of

$$M _ { 3 } ( \mathbb { R } )$$

consisting of all

$$3 \times 3$$

symmetric matrices. Find a set of vectors that spans S.

The given set is a subset of C[−1, 1]. Which of these are also vector spaces? $F=\{f(x)$ in $C[1,1]: f(1)=0\}$

Find the area of the parallelogram whose vertices are listed.

$$(0,0),(-2,4),(6,-5),(4,-1)$$

Express the given vector as a linear combination of the vectors in the given set Q. $f(x)=\cos 2 x$ and $Q=\left\{\sin ^{2} x, \cos ^{2} x\right\}$

Show that W, the set of all 3x3 upper triangular matrices, forms a subspace of all 3x3 matrices. What is the dimension of W? Find a basis for W.

(a) Find the charge, baryon number, strangeness, charm, and bottomness of the $J / \Psi$ particle from its quark composition. (b) Do the same for the $\Upsilon$ particle.

Write the equation of the parabola in standard form, and find the vertex of its graph. y = x$^{2}$ - 8x + 3