Consider the basis S = {v1. V2, V3} for R, where v1 = (1, 1, 1), v2 = (1, 1, 0), and v3 = (1, 0, 0), and let T:R3 → R3 be the linear operator for which T(v1) = (7, – 1, 14), T(v2) = (8, 0, 1), T(v3) = (- 1, 15, 1) Find a formula for T(x1, X2, X3), and use that formula to find T(7, 14, – 1). i i T(7, 14, - 1) = (

Q: 2. Let v, = (1,2,1), v2 = (2,9,0), v3 = (3,3,4). Show first that S = {v1, V2, V3} %3D is a basis for…

A: 1. A finite set of vectors v1,v2,...,vn of a vector space V over a field F is said to be linearly…

Q: Consider the basis S = {v, v2} for R2, where v = (1, 1) and v2 = (1, 0), and let T:R? – R2 be the…

A: Step:-1 Given transformation is  T :  ℝ2 →ℝ2 Tv1 =1, -6  and   Tv2 =-12, 1 Here, v1=1, 1,  v2=1, 0…

Q: 4- Determine whether the vectors v1 = (1, –1, 2), v2 = (3, 2, – 1) and v3 = (5,0, 3) form a linearly…

A:

Q: d) Consider the linearly independent vectors (1, -2,3,-4) and (1, 1, 2, 2) in R¹. Ex- tend {(1,…

A: Consider the linearly independent vectors  and  in .The transformation Now,Represent  as linear…

Q: (-5u^3v^4+9u)+(-5u^3v^4-8u+8u^2v^2)+(-8u^4v^2+8u^3v^4)=

A:

Q: Find the coordinate vector of w relative to the basis S = ({u₁. u₂) for R². u₁ = (4, - 4), u₂ = (2,…

A:

Q: Suppose , t2, ., , are linearly independent. Prove that S is linearly independent where (a) S=…

A:

Q: Consider the basis S = [v;. V2. V3} for Rº, where v = (1, 1, 1), v2 = (1, 1, 0), and v3 = (1,0,0),…

A:

Q: Express the vector t as a linear combination of the vectors u, v and w: t = (5,30, –21), u = (5,2,…

A:

Q: et B= (b,,b2.b3) be a basis for vector, space V. Let T:V V be a linear transformation with the…

A:

Q: Let B = (1, x, x², x*) be an ordered basis for P3. Find the coordinate vector of 8x – 6x2 – 9x + 3…

A:

Q: Q3. a) Express the vector (4,2) e R' as linear combination of the vectors (1,1) and (1,-1). b) Let T…

A: We have to find all these

Q: show that the given vector functions are linearly dependent on (−∞,∞).

A:

Q: Let T R² R² be the linear operator defined by - x2 T([22]) = [ 91 +22] and let B = {u₁, u₂} be the…

A:

Q: Consider the basis S = {v,, v2, V3} for R, where v = (1, 1, 1), v2 = (1, 1, 0), and v3 = (1, 0, 0),…

A: Given that S is basis contains 3 elements S = {v1, v2, v3}, where the basis element is given as:…

Q: If T is defined by T(x)=Ax, find a vector x whose image under T is b, and determine whether x is…

A:

Q: (и, из) for Find the coordinate vector of w relative to the basis S R². Let ui (5, –4), u2 = (3, 7),…

A: Solvig by au+bv=w

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: Consider the ordered basis B = {(1,-1,0), (1, 1, 1), (0, 1, 1)} of R³. the coordinate vector v =…

A: Consider the ordered basis B=1,-1,0, 1,1,1, 0,1,1 of ℝ3.   To Give: The coordinate vector v=-2,-1,0…

Q: 5. Determine whether or not the following vectors i R³ are linearly dependent: (1, –2, 1), (2, 1,…

A:

Q: Let Let T: R² R2 be the linear transformation satisfying Find the image of an arbitrary vector…

A:

Q: 2. Let ū = (-1, 2, 1), ở = (1, – 1,–1) and w = (2, 3, 0). Find the coordinates of (2,4,4) with…

A: Using gauss Jordan elimination method to find the value.

Q: Consider the basis S = {V₁, V₂} for R2, where v₁ = (1, 1) and v₂ = (1, 0), and let T:R² → R² be the…

A:

Q: Consider the basis 5 = {v1, V2, V3} for R°, where v: = (1, 1, 1), v2 = (1, 1, 0), and va = (1,0,0),…

A:

Q: Find the vector ž, which is perpendicular to vectors å = (2,3, –1) and 5 = (1,–2,3) and satisfies…

A:

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: -0-0--0--0--0- 1 19. Let S be a set of vectors such that u = whether the set of vectors is linearly…

A:

Q: Find the coordinate vector of w relative to the basis S = { u1, u2} for R u 1 = (4, – 4), u 2 = (4,…

A: Given: u1=4,-4u2=4,4w=0,24

Q: If a, and b are non - collinear vector and vectors a = (x +4 y)a +(2x + y +1)b and, B= (-2x + y + 2…

A: We will solve the following

Q: Consider the basis S= {V1, V2, V3} for R, where vV1 (1, 1, 1), v2=(1,1,0), and v3 =(1,0,0), and let…

A: Given:       Tv1=8,-1,16        Tv2=9,0,1         Tv3=-1,17,1      v1=1,1,1       v2=1,1,0…