Suppose that ? and ? are subspaces of a vector space ? with ? = ?...

Suppose that ? and ? are subspaces of a vector space ? with ? = ? ⊕ ?. Suppose also that ??, … , ?? is a basis of ? amd ??, … , ?? is a basis of ?. Prove ??, … , ??, ??, … , ?? is a basis of V.

Expert Solution

Colby Messinger answered 1 year ago

Let V be a vector space, and suppose that U and W are both subspaces of...

Let V be a vector space, and suppose that U and W are both subspaces of V. Show that U ∩W := {v | v ∈ U and v ∈ W} is a subspace of V.

Which of the following are subspaces of the vector space of real-valued functions of a real...

Which of the following are subspaces of the vector space of real-valued functions of a real variables? (must select all of the subspaces.) A. The set of even function (f(-x) = f(x) for all numbers x). B. The set of odd functions (f(-x) = -f(x) for all real numbers x). C. The set of functions f such that f(0) = 7 D. The set of functions f such that f(7) = 0

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

Let V be a vector space and let U and W be subspaces of V . Show that the sum U + W = {u + w : u ∈ U and w ∈ W} is a subspace of V .

1- Let W1, W2 be two subspaces of a vector space V . Show that both...

1- Let W1, W2 be two subspaces of a vector space V . Show that both W1 ∩ W2 and W1 +W2 are subspaces.?and Show that W1 ∪ W2 is a subspace only when W1 ⊂ W2 or W2 ⊂ W1. (recall that W1 + W2 = {x + y | x ∈ W1, y ∈ W2}.)

(1) Suppose that V is a vector space and that S = {u,v} is a set...

(1) Suppose that V is a vector space and that S = {u,v} is a set of two vectors in V. Let w=u+v, let x=u+2v, and letT ={w,x} (so thatT is another set of two vectors in V ). (a) Show that if S is linearly independent in V then T is also independent. (Hint: suppose that there is a linear combination of elements of T that is equal to 0. Then ....). (b) Show that if S generates V...

Let {v1, v2, v3} be a basis for a vector space V , and suppose that...

Let {v1, v2, v3} be a basis for a vector space V , and suppose that w = 3v1 − 5v2 + 0v3. For each of the following sets, indicate if it is: a basis for V , a linearly independent set, or a linearly dependent set. (a) {w, v2, v3} (b) {v1, w} (c) {v1, v2, w} (d) {v1, w, v3} (e) {v1, v2, v3, w}

Suppose U is a subspace of a finite dimensional vector space V. Prove there exists an...

Suppose U is a subspace of a finite dimensional vector space V. Prove there exists an operator T on V such that the null space of T is U.

suppose that T : V → V is a linear map on a finite-dimensional vector space...

suppose that T : V → V is a linear map on a finite-dimensional vector space V such that dim range T = dim range T2. Show that V = range T ⊕null T. (Hint: Show that null T = null T2, null T ∩ range T = {0}, and apply the fundamental theorem of linear maps.)

(3) Let V be a vector space over a field F. Suppose that a ? F,...

(3) Let V be a vector space over a field F. Suppose that a ? F, v ? V and av = 0. Prove that a = 0 or v = 0. (4) Prove that for any field F, F is a vector space over F. (5) Prove that the set V = {a0 + a1x + a2x 2 + a3x 3 | a0, a1, a2, a3 ? R} of polynomials of degree ? 3 is a vector space over...

Let T be an operator on a finite-dimensional complex vector space V, and suppose that dim...

Let T be an operator on a finite-dimensional complex vector space V, and suppose that dim Null T = 3, dimNullT2 =6. Prove that T does not have a square root; i.e. there does not exist any S ∈ L (V) such that S2 = T.

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