4.35 When we decompose a vector space as a direct sum, the dimensions of the subspaces add to the dimension of the space. The situation with a space that is given as the sum of its subspaces is not as simple. This exercise considers the two-subspace special case. (a) For these subspaces of M2x2 find W₁ W₂, dim(W₁ W₂), W₁ + W₂, and dim(W₁ + W₂). W₁ = {(d) | c, d = R} W₂ = {(b) | b, c € R} (b) Suppose that U and W are subspaces of a vector space. Suppose that the sequence (B1,..., Bk) is a basis for Un W. Finally, suppose that the prior sequence has been expanded to give a sequence (μ₁,..., ₁, B₁,..., ßk) that is a basis for U, and a sequence (B₁,..., Bk, w₁,..., wp) that is a basis for W. Prove that this sequence (μ₁, ..., µj, B1, ..., Bk, w1,..., p) is a basis for the sum U + W. (c) Conclude that dim(U+W) = dim(U) + dim(W) - dim(Unw). (d) Let W₁ and W₂ be eight-dimensional subspaces of a ten-dimensional space. List all values possible for dim(W₁ W₂).

4.35 When we decompose a vector space as a direct sum, the dimensions of the subspaces add to the dimension of the space. The situation with a space that is given as the sum of its subspaces is not as simple. This exercise considers the two-subspace special case. (a) For these subspaces of M2x2 find W₁ W₂, dim(W₁ W₂), W₁ + W₂, and dim(W₁ + W₂). W₁ = {(d) | c, d = R} W₂ = {(b) | b, c € R} (b) Suppose that U and W are subspaces of a vector space. Suppose that the sequence (B1,..., Bk) is a basis for Un W. Finally, suppose that the prior sequence has been expanded to give a sequence (μ₁,..., ₁, B₁,..., ßk) that is a basis for U, and a sequence (B₁,..., Bk, w₁,..., wp) that is a basis for W. Prove that this sequence (μ₁, ..., µj, B1, ..., Bk, w1,..., p) is a basis for the sum U + W. (c) Conclude that dim(U+W) = dim(U) + dim(W) - dim(Unw). (d) Let W₁ and W₂ be eight-dimensional subspaces of a ten-dimensional space. List all values possible for dim(W₁ W₂).

Elementary Linear Algebra (MindTap Course List)

8th Edition

ISBN:9781305658004

Author:Ron Larson

Publisher:Ron Larson

Chapter5: Inner Product Spaces

Section5.CR: Review Exercises

Problem 41CR: Let B={(0,2,2),(1,0,2)} be a basis for a subspace of R3, and consider x=(1,4,2), a vector in the...

See similar textbooks

Similar questions

Let B={(0,2,2),(1,0,2)} be a basis for a subspace of R3, and consider x=(1,4,2), a vector in the subspace. a Write x as a linear combination of the vectors in B.That is, find the coordinates of x relative to B. b Apply the Gram-Schmidt orthonormalization process to transform B into an orthonormal set B. c Write x as a linear combination of the vectors in B.That is, find the coordinates of x relative to B.
Let A be an mn matrix where mn whose rank is r. a What is the largest value r can be? b How many vectors are in a basis for the row space of A? c How many vectors are in a basis for the column space of A? d Which vector space Rk has the row space as a subspace? e Which vector space Rk has the column space as a subspace?
Find an orthonormal basis for the subspace of Euclidean 3 space below. W={(x1,x2,x3):x1+x2+x3=0}
Suppose that S1 and S2 are subspaces of a vector space (V, F). Show that their intersection S1 ∩ S2 is also a subspace of (V, F). Is their union S1 ∪ S2 always a subspace?
4. Consider the following subspaces of P. H = Span{1+t, 1-t3 } and G = Span{1+t+t 2, t – t 3, 1+t+t 3} Find dim H, dim G and dim H intersection G.
10. Find a basis for the Subspace. What is the dimension of W?
If S and T are subspaces of V, then SintersectionT is a subspace. Select one: True False
Consider the following subspaces of R3: U=span[(1, 2, 1), (1, 3, 0)] W=span[(1, 1, -1), (-1, -1, 1)] Sub-Task (1). Find a homogeneous systems whose solution space is U. Sub-Task (2). Find a homogeneous systems whose solution space is W. Sub-Task (3). Find the dimension of UW.
Is the set M={1/n | n ε z+} compact as a subspace of R?
Determine whether the following are subspaces of C[−1, 1]: The set of continuous nondecreasing functions on [−1, 1]
How do you prove that W= im T when W is a T-invariant subspace, and V=ker + W. Where V is finite dimentional, when you let T be a element V.
4. Consider the following subspaces of P.H = Span{1 + t, 1 − t3} and G = Span{1 + t + t2, t − t3, 1 + t + t3}Find dim H, dim G and dim H ∩ G.