(a) Let S and T be subspaces of a vector space V. Show that SUT is again a subspace of V if and only if SCT or T C S. Hence show that if S and T are both proper subspaces of V then SUT+V.

Q: 2. Let V be a vector space over R, and let S1 and S2 be subspaces of V. (a) Prove that Si C S2 =…

A: I have solved by assuming basis of the givem subspaces

Q: (b) If V = R", and U is a subspace of V of dimension m with 0 <m <n, prove that U² = {v € V: B(u, v)…

A: As per our guidelines we are supposed to solve only one problem. For the solution of the first…

Q: 2. Let U and W be subspaces of a vector space V such that UUW is a subspace of V. Prove that UCW or…

A: Given that U and W are subspaces of a vector space V such that union of U and W is a subspace of V.…

Q: 6. Let V be a vector space and let U1 and U2 be subspaces of V. Show that if U1 U U2 is a subspace…

A:

Q: Let P2 be the vector space of all polynomials of degree 2 or less, and let H be the subspace spanned…

A: We first write given vectors is matrix form and then reduce the given matrix in row echelon form.…

Q: Let U and W be subspaces of V over a ficld F. Show that the union U UW is a subspace if and only if…

A:

Q: Let A = { | 2.x |x € R}. Is A a subspace of R³? З Let B x + y + z = 0}. Is B a subspace of R³?

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Q: 2. Let H be the set of all vectors of the form (a – 3b, b – a, a, b) where a and b are arbitrary…

A: Let V be a vector space. H is said to be a subspace of V. If  H is a subset of V. H is closed with…

Q: Let S be a subspace of R" and let S be its orthogonal complement. Prove that S- is also a subspace…

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Q: Let V be a vector space, F a collection of subspaces of V with the following property: If X,Y EF.…

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Q: Let V be a vector space of dimension 8. How many subspaces out of V can you make? A. 2 subspaces B.…

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Q: If S and T are subspaces of a vector space V, then SnTis a subspace of V. Select one: a. a subspace…

A: We have to prove that if S and T are subspace of a vector space V, then S∩T is a subspace of V or…

Q: True or False: If U and W are subspaces of a finite dimensional vector space and UnW = {0}, then U°…

A:

Q: Q. 5 If V is any vector space, show that (0} and V are subspaces of V.

A: We have that V is a vector space.  Now, V is said to be a vector space over a field F along with 2…

Q: 6. Let U and W be 3-dimensional subspaces of a 4-dimensional vector space V, and suppose that U # W.…

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Q: Let V be a vector space with subspaces U and W. Define the sum of U and W to be U + W = { u + w : u…

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Q: Given subspaces H and K of a vector space V, the sum of H and K, written as H + K, is the set of all…

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Q: 1. Show that WW10 W2 is a vector subspace of V, but that W1 U W2 need not be a vector subspace in…

A: according to our guidelines we can answer only three subparts, or first question and rest can be…

Q: Show that if Z subspace of a space Y and Y is a subspace of a space X then Z is a subspace of a…

A:

Q: If S and T are subspaces of V, then SUT is a subspace. Select one: True False

A: Given S and T are two subspacewe have to check whether S∪T  is subspace or notClearly  S∪T  is…

Q: 6. (a) Determine whether the set of all vectors of the form (a,b,c), where b a+c, is a subspace of…

A: The subspace of R3 is (a,b,c) where b=a+c

Q: If S is a subspace in R that is spanned by (:) G)-G) then S- is spanned by: а. 1 and 1 1 1 and -1 С.…

A:

Q: If W is a subspace of R and v is any vector in R", then ||projwv|+|v-projwv||2= |v||2.

A: Introduction: Every vector that doesn't lie in a subspace can be uniquely expressed as a sum of two…

Q: 2. Let V = E3 and W = {x|x = a(1,0,2) + b(1,-1,3)} and S = {x\x = (x1,x2, 1)} where a and b are any…

A:

Q: (b) Let V be an n-dimensional F-vector space, and let 0 < k < n. If K is a k- dimensional subspace…

A:

Q: True or False? No justification is required. (a) If U is a subspace of a vector space V, then U'…

A: Consider the provided question, Hello. Since your question has multiple sub-parts, we will solve…

Q: 17. Prove that a subset W of a vector space V is a subspace of V if and only if W + Ø, and, whenever…

A: We Know that Let V be a vector space over the field F and let W E V . Then W will be a subspace of V…

Q: a Let W ER a + b+ c = 0 Prove that W is a subspace of R Find a basis for W and state the dimension…

A:

Q: 6. (a) Suppose U, W are subspaces of V. Show that (U + W)° = U°nW°. (b) Suppose V is…

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Q: W= im T when W is a T-invariant subspace, and V=ker + W. Where V is finite dimentional, when you let…

A:

Q: snip

A: We can represent the given set again as: W=xyz:x=3y+1 and z=-2y Let  w1=x1y1z1∈WThen,x1=3y1+1 and…

Q: Prove that a subset W of a vector space V is a subspace of V if and only if 0 EW and ax+ y E W…

A: First proving the first part: If W is the subset of V and also the subspace of V Claim  : 0 W and…

Q: 3) Find a basis for the subspace of R that consists of the intersection of the planes X-3Y = 5Z and…

A: This is a problem of vector space.

Q: 5. Let S = {x + 1,x2 – 2, x – 1,3}. Find a basis for the subspace W that is spanned by S.

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Q: Let H and G be subspaces of a vector space V . Show that H \ G is a subspace of V . (Note that the…

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Q: In R, let S be the set of all vectors with third component 0. { a S = ER Show that S is a subspace…

A:

Q: 5 Let u,= Note that u, and u, are orthogonal. It can be shown that u, is not in the subspace W…

A: Given - Let u1 = 44-8, u2 = 5-12 and u3 = 010. Note that u1 and u2 are orthogonal. It can be shown…

Q: 3) Show that the set of all elements of the form (x + 2y,x + 3z, 2x + y + z), where x, y, z E F, is…

A: We have, (x+2y , x+3z , 2x+y+z),where x, y, z € F.

Q: Let V=R^2 and let H be the subset of V of all points on the line 2x+3y=6. Is H a subspace of the…

A: Solution:Given :     V=ℝ2and      H= (x,y)/ 2x+3y=6   ⊂ ℝ2

Q: Please solve this question. Thank you

A:

Q: 43. (a) If W is a subspace of a vector space V, then it has closure under scalar multiplication as…

A:

Q: 1. Decide if the following subsets W C V are vector subspaces or not (justify) (a) W = x €RCR?. COs…

A:

Q: 3. Let V be a vector space over R with subspaces U and W. (a) Show that U nW is a subspace of V. (b)…

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Q: 4) Suppose u, v E V, where V is an inner product space, and ||u|| = |||| = 1 and (u, v) = 1. Find u…

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Q: Let V and W be vector spaces with subspaces V1 and W1, respectively. If T : V --&gt; W is linear,…

A: Given: T:V →W is a linear transformation, V1, W1 are respectively subspaces of V and W

Q: Suppose W1 and W2 are subspaces of a vector space, V. Define W as follows: W = {w1 + W2 | W1 E W1…

A: As W1 is a subspace of V , hence (a) 0∈W1 (b) αw1+βw1'∈W1 for every w1 , w1'∈W1 As W2 is a subspace…

Q: 18. A vector space V is said to be the direct sum of its subspaces U and W, written V = U®W, if…

A: (a)  Given, V=U⊕W and prove that every vector in V is the sum of some vector in U and some vector in…

Q: 5 Let v, = 0 . v, = and w= 1 Is w in the subspace spanned by {v,. V2. V3)? Why? Choose the correct…

A: Given: v1=10-1, v2=415, v3=929 and w=516

Q: 3. Let H andG be subspaces of a vector space V. Show that HNG is a subspace of V. (Note that the…

A:

Q: 1. Let V be a finite-dimensional inner product space and let wj, 10 vectors in V (with absolutely no…

A: Given that U=v∈V:(v,wj)=0,1≤j≤k (a) The aim is to show that U is a subspace of V. We say U is…

Linear Algebra

In mathematics, problems can be solved by the arithmetic operators like addition, subtraction, multiplication, and division. In algebra, we represent the numbers by any letter called a variable. If these variables are of degree 1, then it is studied under linear algebra. The first-degree equations involving algebraic expressions are called linear equations. That is, if you represent it by drawing a graph you will get a straight line.