Let V be a vector space and T: V->V be a linear map such that T^2=T. 1) Prove that lm(T)=ker(Id-T) 2) Prove that V=ker(T)⊕Im(T)

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Q: (i) Consider the vector space M2×2 with the trace inner product ( A,B) = trace(A'B) and let A = .…

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Q: let W be an inner product space and let w1 and w2 be vectors in W. suppose ll w1 ll =5^1/2 and ll w2…

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Q: Let W1=span{(1,1,0),(-1,1,0)} and W2=span{(1,0,2),(-1,0,4)} Proove W1+W2=R^3

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Q: Let V be a finit dimensions vector space of at least dim =1, Let T: V-> V be a linear transformation…

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Q: 2s Let H be the set of all vectors of the form Find a vector v in R' such that H span {i}. -s S

A: Spanning set of H

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Q: Let V be a finit dimensions vector space of at least dim =1, Let T: V-> V be a linear transformation…

A: Given that V be a finite dimensional vector space with dimension ≥1 and T: V→V be a linear…

Q: iii) (f x g) xh is differentiable at t and d dt 7 {(f x g) x h} xgx h + x dt dg xh +(fx g) x dt dt

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Q: Let U₁ = (U1, V2) (U1, U₁) V1 = U₂ || (U1, V3) (U2, V3) Use the Gram-Schmidt procedure to produce an…

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Q: Let V = R' and define vector addition as (x, y, z)+(x', y', z')= (x+x',y+y',z+z') (standard…

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Q: Describe all linear maps ƒ : R² → R² that are projections with image the subspace ((1,2)).

A: f: ℝ2→ℝ2Subspace =1, 2

Q: 3. Find a parametrization for (a vector valued function that traces) the line which passes through P…

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Q: Verify all vector space axioms under the given operations (x,y,z)+(x',y'z') = (x+x'+1,y+y'+2,z+z'+3)…

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Q: Use the function to find the image of v and the preimage of w. T(Vg, V2) = (V, + V2 V1 – V2), v =…

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Q: Let $K$ is a horizontal shear, next draw a vector $\vec{x}=\left[\begin{array}{l}x_1 \\…

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Q: 2. Let S = span{V₁, V2, V3}. Check whether the vector u € St. (a) v₁ = [1, 0, 0, 0], v2 = [0, 3, 0,…

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Q: Use the function to find the image of v and the preimage of w. T(V1, v2) = (Vị + v2, V1 - V2), v =…

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Q: Find a vector-valued function whose graph is the plane x+y+z= 12, O r(u, v)=-ui-vj+(12-u-v)k O r(u,…

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Q: Let f,g be scalar functions, ủ be a vector function, and assuming sufficient differentiability. Then…

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