Let X,Y be the two vector spaces over the same field F and T:X-Y is a function such that T(ax+by) = aT(x)+bT(y). Then (i) R(T) = T(X) = {T(x): x€ X} is subspace of Y. (1) N(T)- Ker (T)-{x € X: T(x) -0 } is a subspace of X.

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Q: Let X,Y be the two vector spaces over the same field F and T:X_Y is a function such that. i) R(T)=…

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Q: 3) Suppose TE L(V) and U1, . , Um are subspaces of V invariant under T. Prove that U, ...+ Um is…

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A: Let S={v1,v2,v3} represent the orthonormal basis for the subspace. By Gram Schmidt Process,…

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A: Our Aim is to show that the set W of all solutions of the differential equation3d2ydx2-5dydx+7y=0…

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Q: 10. Is 2 = {(x, y,z)| x=2 y & z=1} a subspace of R³?

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Q: Let W = {a + bx + cx + dx| a+ b = 0 and c – 3d = 0} be a subspace of P3 Then the dimension of W is…

A: Linear combination of vectors we use to find.

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Q: 10. Let Y = {(x,y,z)| z=-x}. (a) Is Y a subspace of R³? Justify. (b) Is Y a vector space? Justify.

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Q: Find the orthogonal projection ŷ of y = onto the subspace W = Span { uị u2 5 Ex: 1.23 : ŷ =

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Q: et X = {(x,y, z): Vx² + y² + z² <1} be a subspace R° with the standard topology and 0 = (0,0,0).…

A: Given,               X=x, y, z:x2+y2+z2 ≤1  be a subspace of R3 with the standard topology and o=0,…

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Q: Let V = P2(t) with inner product defined by (f, g) = So f(t)g(t) dt. Find a basis for the subspace W…

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Q: 26. In the vector space of all real-valued functions, find a basis for the subspace spanned by {sin…

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Q: Suppose V is finite-dimensional and U and W are subspaces of V with W^0 ⊂ U^0. Prove that U ⊂ W.

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Q: if a vector space is the set of real valued continuous functions over R , then show that the set W…

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A: Subspace

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Q: Suppose that U cr|xyer F4 y EF, U is a subspace of F4- you don't have to! Find a subspace V of F4…

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Q: Suppose U = {(x, y,x + y,x – y, 2x) e F° : x, y e F}. Find a subspace W of F such that F5 = U + W.

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Q: 1) Suppose that V is a finite dimensional vector space over R. Prove that every T E L(V) has an…

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Q: Give a counter example to show that W is not subspace of R³. W=set of (x, y, z) where x + y = 1 6. ,

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Q: a) T: M, → R, where M, is the vector space ofn by n matrices over a field, and T (A) = det(A). b)…

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Q: 5. Let C' (R) be the set of all continuously differentiable functions. Show that W = {f : f' = 4f}…

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Q: 12. Let K C R" and let V = {x € R" : x ly Vy E K}. Prove that V is a subspace of R".

A: " As per the company guidelines we are supposed to solve first question only. Kindly post the other…

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Q: Example 3. The operator Ax = d/dt x(t), defined on the subspace M of C[0, 1] consisting of all…

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Q: a) T: M, → R, where M,n is the vector space of n by n matrices over a field, and T (A) = det(A). b)…

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Q: Let B = {1, x, ex, xex} be a basis for a subspace W of the space of continuous functions, and let Dx…

A: Given: B=1, x, ex, x ex is a basis g=for a subspace W of the space of continuous functions. DX is…

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Q: Let W = (a + bx + cx² + dx³[ c – 3d = 0 } be a subspace of Py. Then the dimension of W is equal to

A: Here we will find out dimension of W.

Q: Prove that S = {(x, y, z) E R³: 2x + 3y+z = 0} is a subspace of the vector space R3.

A: Explanation of the answer is as follows

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Q: a) T: M, → R, where M, is the vector space ofn by n matrices over a field, and T (A) = det(A). b)…

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