Let U=span(ū, ,ū,,ū and Û=span {u, ,u,, where ü,=(1,0,0,0), ū,=(0,1,1,0), ủ,=(0,1,1,1) and ů,=(1,0,0,1), ủ,=(1,1,0,0), ủ=(0,0,1,1). Show that U & Û are vector spaces (over the field R).

Q: Let U=span u, ,ū2, ūz and Ü=span{ú, ,ú2,úz where ü,=(1,0,0,0), ủ,=(0,1,1,0), ủ,=(0,1,1,1) and…

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Q: State if the following statement is true or false: If y is a vector space of dimension , then any…

A: Given:

Q: › affine space (E‚È‚+), let (ai)ie is linearly independent for some i

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Q: Let U=span{uq ,U2,U3} Û=span {u, ,u,,u,} where ū,=(1,0,0,0), ū,=(0,1,1,0), ū,=(0,1,1,1) and…

A: (a) Given vectors are u1^→=1,0,0,1,u2^→=1,1,0,0,u3^→=0,0,1,1.   Consider U^=spanu1^→, u2^→, u3^→.…

Q: Let V is finite vector space and suppose that V=internal direct sum of U and S1. Also V=internal…

A: Either S1 and S2 will be disjoint or identically same. If S1 is a subset of S2 then S1= S2.

Q: 15- Let A C B C V,and let V be a vector space, then if A is linearly dependent, then B is A:…

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Q: Let ((u,,u2,U3),(V1,V2,V3))= 2u, V1 + 3u2V2+ U3V3 be an inner product on R. Determine ||(1, – 1, -…

A: Explanation of the answer is as follows

Q: If V is vector space, with u = (u, , U2) and v = (v, , v2) with u +v = (u,+v, , u2+v2) and ku =…

A: u=(u1,u2) , ku=(3u1,0) k=2, u=(-1,2)

Q: Let U = span{(1,1, 1), (1, 1, 0)} C R°, V = span{(1,2, 0), (1, –1,0)} C R°, and let W = span{(1, 1,…

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Q: Find a subset of u1, u2, u3, u4 that gives a basis for W=Span{ui} of R5 , where a) u1= (1,0,1,0,1),…

A: Answer

Q: 15- Let A CB V,and let V be a vector space, then if A is linearly dependent, then B is A: linearly…

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Q: Let W = (0, 1, 3}, X = {1, 3, 5), Y = (3, 4} and Z = {(1, 2), (3, 4), (1, 3),(2, 4). Compute (X x Y)…

A: The given problem is to find the Cartesian product of given sets and verify the Cartesian product…

Q: Suppose v1, v2, v3, v4 span a vector space ?. Prove that the list v1 − v2, v2 − v3, v3 − v4, v4 also…

A: Let V be the vector space Since v1,v2,v3,v4 span the vector space V.

Q: Let V, W, Z be vector spaces and let T:V-->W and U:W-->Z be linear transformations. Show that if T…

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Q: Let F, G be linear maps of vector space V onto itself. Show that: (FoG)-1 = G¬1 oF-1.

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Q: Let u=(6,2,4) and v=( 1,2,0). Find projection of u onto v.

A: topic - projection of vectors

Q: linear algebra question.

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Q: Show that UE span{(1,2, – 1,0),(1,1,0,1),(0,0, – 1,1)} where u= (2,5, – 5,1) by finding scalars k,I…

A: We have,u=2,5,-5,1andu=k1,2,-1,0+l1,1,0,1+m0,0,-1,1---1

Q: 19. Prove that the set {(1, 0, 0), (1, 1, 0), (1, 1, 1), (0, 1, 0)} spans the vector R' (R) but is…

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Q: Define spanning set. Determine whether the vectors u = (1,2,0), 2 = (-1,1,2) and uz = (3,0,–4) span…

A: We need to define spanning set and to determine if the given vectors span R3  This can be done as…

Q: Let F,G be linear maps of vector space V onto itself. Show that: (FoG) = G -1 F-1,

A: Given that F,G be linear maps of vector space V onto itself.

Q: Show that 4(u, v) = ||u+v||² – ||u – v||2 for all u, v in an inner product space V (V - -

A: Given: The given equation is, 4u,v = u+v2-u-v2 Here we know, real vector space is V so that u,v is…

Q: 2. Let S, T,U be linear transformations such that (letting {u1, u2} or {u, , u2, uz} be bases of the…

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Q: 5 (а) Show that if we think of C as a vector space over R, then the list (1+ i,1 – i) is linearly…

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Q: 5.2. Prove that if V is a vector space having dimension n, then a system of vectors V1, V2, ..., Vn…

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Q: 3. Let F,G be linear maps of vector space V onto itself. Show that: (FoG)-1 = G-1o F-1.

A: To prove F∘G-1=G-1∘F-1, Where F and G are linear maps of vector space.

Q: Let ((uz,u2,U3),(V1,V2,V))= 2U1V1 + 3u2V2+ U3V3 be an inner product on R, Determine d(1, – 1, –…

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Q: 5. Let V be an inner product space over C. Show that for any v, w ɛ V ||u||² + ||v||² (v, (v, w)w) =…

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Q: Let U = span((1, 1, −3, −3),(2, −2, −2, 2)) and W = span((1, 0, 2),(3, 1, 1),(9, 4, −2),(−7, −3,…

A: Yes. U and W both are isomorphic to each other. Because the dimension of the spanning set of U is 2…

Q: the topology 7, = {X,ø{a},{b,c}} 1; = {Y.ø.{u}} B={X ×Y,6.{(a,u}…

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Q: B\ If X is Pre-Hilbert space then (x,ŻB,Y, ) = ΣP, (x,y, (x, j=1 j=1

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Q: x1 ,x2 ,...,xn  span a vector space V, then they are linearly independent. Select one: True False

A: We have to check

Q: Let V be an inner product space. Show that if w is orthogonal to each of the vectors u1, U2, ·..,…

A: Inner product space: Let V be a vector space , then  v is said to be inner product space if  it…

Q: Prove that ||u+v||2 + ||u-v||2= 2||u||+2||v|| for any u and v in an inner product space V

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Q: Let R = {(0,1), (0,2), (1,1), (1,3), (2,2), (3,0)} Find R', the transitive closure of R

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Q: Define spanning set. Determine whether the vectors u Uz = (3,0, –4) span R³. 2. (1,2,0), u2 =…

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Q: 4. Let U=span{ū, , ū,, ū;} and U=span{û ,û, ûz} where ū,=(1,0,0,0), ū,=(0,1,1,0), ū,=(0,1,1,1) and…

A: U=spanu1→,u2→,u3→ and U^=spanu1^→, u2^→, u3^→where,u1→=1,0,0,0, u2→=0,1,1,0, u3→=0,1,1,1…

Q: Let U=span{ū, ,ū,,ūz and Û=span{u ,u,,u where ū,=(1,0,0,0), ū,=(0,1,1,0), ū,=(0,1,1,1) and…

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Q: Define spanning set. Determine whether the vectors u1 = (1,2,1), u2 = (−1,1,2) and u3 = (2,0, −4)…

A: SPANNING SET: If S=v1,v2,v3,............,vr is a set of vector space V. Then the set of all linear…

Q: Let (x,r, x,x, 1} is the basis of the set P. then the dimension of P is

A: We know, Dimension is the number of elements in the basis of the set.

Q: 5. Let U=span{ū ,ū2,ū;} and Ü=span{u ,ú,úz where ū,=(1,0,0,0), ū,=(0,1,1,0), ū,=(0,1,1,1) and…

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Q: 3. Show that if two vectors in the set {a1, a2,... , ak} are identical, then the set is linearly…

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Q: If V (F) is a finite dimensional vector space of dimension n then each set of (n+ 1) or more vectors…

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Q: Show that if {Po, P₁, Pn} is an orthogonal set of functions on [a, b] with respect to the weight…

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Q: Let В %3 (х, у, z) be a basis for vector space V, and let S = {x, y + z, –x} What is the dimension…

A: To find: the dimension of Span(S). where S={x,y+z,-x}. and B={x,y,z} is a basis for a vector space…

Q: If R = {(1, 2), (1, 4), (2, 3), (3, 1), (4, 2)}, what is the symmetric closure of R? %3D О (1, 2),…

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Q: Find a subset of u1, uz, uz, U̟ that gives a basis for W = span(u;) of R°, where %3D (а)) иј 3D %3D…

A: Given that u1=1,1,1,2,3 u2=1,2,-1,-2,1u3=3,5,-1,-2,5u4=1,2,1,-1,4 find a subset

Q: Show that (5,0,3) is in the span of S= {(1,1,0), (1,0,1), (1,1,1)}.

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Q: Let V be inner product space, let u, v E V, then one of the following cannot be true A) || u ||= 3,…

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Q: 1 Exercise 5.1.4 Let u = V = 3 and w = Show that span {u,v, w} = span {u, v}. 5

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