Let H= {p(t): p(t)= a+ bt+ ct°; a,b,c ER } (a) Show that H is a subspace of P3: (b) Let p 1, P2, P3 be polynomials in H, such that p,(t) = 2, p2(t)=1+3f° , p3(t)= – 1-t-8. Use coordinate vectors in each of the following andjustify your answer each part. (i) Verify that {p1, P2, P3} form a linearly independent set in P3 (ii) Verify that {p, P2, P3} does not span P3. (iii) Can the set {p,, pɔ, p3} form basis for P3? 11 (c) Let T:H→ R* be a linear transformation, defined by p(0) - p(0) T(p) = p(0) Lp(0) Find Ker T and find two polynomials that span Ker T.

Q: Prove that the vectors (1,1,0), (3,1,3) and (5,3,3) are linearly dependent.

A:

Q: Let T: R² → R² be a reflection in the x-axis. Find the image of each vector. (a) (2,7) (b) (6, -8)…

A: T: R2 - >  R2 is a inear map which is basically reflection in the x Axis. Therefore for any point…

Q: Let ū = (4, –2), ū = (-2,0), and w = (2, 2). Find the vector x that satisfies Tu – u +x = 5x + w. In…

A: Given Let Plugging the values in the equation:

Q: Q2) Find the coordinate vector [x]B of x relative to the basis B={b1, b2}:

A: Let us consider a vector V  with the basis b1,b2,......bn then any vector v∈V is written as linear…

Q: Let u = (-1,0,1), v= (3,2,1) and w= (0,1,2). (b) Prove that the vectors u v and w are linearly…

A:

Q: the function to find the image of v and the preimage of w. T(V1, V2, V3) = (4v2 - V1, 4V, + 5v), v =…

A:

Q: Find the projection of vector u⃗ = −i + 5j + 3k onto the vector v = −i + j − 3k?

A: Projection of vector u on vector v is given by  Projv(u) = u.vv2 v Here u = -i+5j+3kv = -i+j-3k…

Q: Use the function to find the image of v and the preimage of w. V2, V1 + V2, 2v, – v2 2 v = (5, 5), w…

A: Here we have a function,we have to find image of v and pre image of w.

Q: Find a vector function that represents the plane that passes through thepoint p0 with position…

A: The plane passes through the point p0 whose position vector is r0→. The plane also passes through…

Q: Use the function to find the image of v and the preimage of w. T(V1, V2, V3) = (v2 - V1, Vị + V2,…

A:

Q: Let y = *₁ = 0 and ū= 6 . Write y as the sum of two orthogonal vectors, ₁ in Span{u} and 2…

A: Given that the vectors  y→=0-1-2   and  u→=56-1   .

Q: Let 13 20 x 20+ y = 2 be the equation of a curve in the xy-plane. Find the equation y = T(x) of the…

A:

Q: GeoGebrä Task 1: This rectangle has been decomposed along its diagonal. Recompose the two pieces to…

A: A rectangle is a closed quadrilateral with four sides and four angles. All the four angles of the…

Q: If L is a line in 2-space or 3-space that passes through the points A and B, then the distance from…

A: Answer is 10.1

Q: Let T: R2 R³ and u, v E R2 such that T(u) = 1 2 -2 and T(v) = 4 -2 1 Find the image under T of the…

A:

Q: Consider the set of vectors B = {(1, 0, 2), (2, 1, -1), (3, 0, 7)} (1) Is 3 linearly independent?…

A:

Q: Find a linear mapping G that maps [0, 1] × [0, 1] to the parallelogram in the xy-plane spanned by…

A: The linear map G maps  to the parallelogram in XY-plane.The parallelogram is spanned by the vectors…

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.