1. Let U = {r, s, t, u, v, w, x, y, z}, D = {s,...

1. Let U = {r, s, t, u, v, w, x, y, z}, D = {s, t, u, v, w}, E = {v, w, x}, and F = {t, u}. Use roster notation to list the elements of D ∩ E.

a.	{v, w}
	b.	{r, s, t, u, v, w, x, y, z}
	c.	{s, t, u}
	d.	{s, t, u, v, w, x, y, z}

2. Let U = {r, s, t, u, v, w, x, y, z}, D = {s, t, u, v, w}, E = {v, w, x}, and F = {t, u}. Use roster notation to list the elements of D ꓴ E.

	a.	{v, w}
	b.	{r, s, t, u, v, w, x, y, z}
	c.	{s, t, u}
	d.	{s, t, u, v, w, x}

3. Let U = {r, s, t, u, v, w, x, y, z}, D = {s, t, u, v, w}, E = {v, w, x}, and F = {t, u}. Use roster notation to list the elements of E ∩ F.

a.	{t, u, v, w, x}
	b.	{t, u, v, w, x}
	c.	{Ø}
	d.	Ø

4. Let U = {r, s, t, u, v, w, x, y, z}, D = {s, t, u, v, w}, E = {v, w, x}, and F = {t, u}. Use roster notation to list the elements of (E ꓴ F)' ∩ D.

	a.	Ø
	b.	{t, u, v, w, x, z}
	c.	{s}
	d.	{r, t, u, v, w, x, y, z}

Expert Solution

milcah answered 2 years ago

3.5.4 ([Ber14, Ex. 3.6.14]). Let T : V → W and S : W → U...

3.5.4 ([Ber14, Ex. 3.6.14]). Let T : V → W and S : W → U be linear maps, with V finite dimensional. (a) If S is injective, then Ker ST = Ker T and rank(ST) = rank(T). (b) If T is surjective, then Im ST = Im S and null(ST) − null(S) = dim V − dim W

Let X ∈ L(U, V ) and Y ∈ L(V, W). You may assume that V...

Let X ∈ L(U, V ) and Y ∈ L(V, W). You may assume that V is finite-dimensional. 1)Prove that dim(range Y) ≤ min(dim V, dim W). Explain the corresponding result for matrices in terms of rank 2) If dim(range Y) = dim V, what can you conclude of Y? Give some explanation 3) If dim(range Y) = dim W, what can you conclude of Y? Give some explanation

Consider the linear transformation T: R^4 to R^3 defined by T(x, y, z, w) = (x...

Consider the linear transformation T: R^4 to R^3 defined by T(x, y, z, w) = (x +2y +z, 2x +2y +3z +w, x +4y +2w) a) Find the dimension and basis for Im T (the image of T) b) Find the dimension and basis for Ker ( the Kernel of T) c) Does the vector v= (2,3,5) belong to Im T? Justify the answer. d) Does the vector v= (12,-3,-6,0) belong to Ker? Justify the answer.

1. Let V and W be vector spaces over R. a) Show that if T: V...

1. Let V and W be vector spaces over R. a) Show that if T: V → W and S : V → W are both linear transformations, then the map S + T : V → W given by (S + T)(v) = S(v) + T(v) is also a linear transformation. b) Show that if R: V → W is a linear transformation and λ ∈ R, then the map λR: V → W is given by (λR)(v) =...

x, y, z, w, u and t are integer variables, what will their value be after...

x, y, z, w, u and t are integer variables, what will their value be after the execution of the statements below. Give the final value of each variable x = 7; y = x + 1; z = x % (y – 2) + 4; y = (y + z ) % (x + 4) ; w = (x * y) / (z – 3); x = x + x; u = w - 3; t = z +...

Let A = R x R, and let a relation S be defined as: “(x1, y1)...

Let A = R x R, and let a relation S be defined as: “(x1, y1) S (x2, y2) ⬄ points (x1, y1) and (x2, y2)are 5 units apart.” Determine whether S is reflexive, symmetric, or transitive. If the answer is “yes,” give a justification (full proof is not needed); if the answer is “no” you must give a counterexample.

Question

1. Let U = {r, s, t, u, v, w, x, y, z}, D = {s,...

Solutions

Expert Solution

Related Solutions

3.5.4 ([Ber14, Ex. 3.6.14]). Let T : V → W and S : W → U...

Let X ∈ L(U, V ) and Y ∈ L(V, W). You may assume that V...

Consider the linear transformation T: R^4 to R^3 defined by T(x, y, z, w) = (x...

1. Let V and W be vector spaces over R. a) Show that if T: V...

x, y, z, w, u and t are integer variables, what will their value be after...

Let A = R x R, and let a relation S be defined as: “(x1, y1)...

Given r(t)=ti+2sintj+2costk and u(t)=1/ti+2sintj+2costk, find the following: 1. r(t) x u(t) 2. d/dt (r(t) x u(t)...

Let V be a vector space and let U and W be subspaces of V ....

If z=x2y- 9xy2 and x=u2+euw, y=v2+2uv, when u=1, v=-1, w=0, then Zu=

Let X and Y be random variable follow uniform U[0, 1]. Let Z = X to...