3. (a) Let S = {v1, . . . , vn} be a linearly dependent subset of a vector space V , and let vn+1 be an element of V which is not an element of S. Prove/disprove: S∪{vn+1} is linearly dependent. (b) Let T = {v1, . . . , vn} be a linearly independent subset of a vector space V , and let vn+1 ∈ V \ T. Prove that T ∪ {vn+1} is linearly dependent if and only if vn+1 ∈ span(T)

Q: If (x1,y1) + (x2,y2) = (x1x2,y1y2) and  c(x1,y1)= (cx1,cy1) are vector spaces, verify if both…

A:

Q: Suppose ....,V4}is a linearly dependent spanning set for a vector space V. Show that each w in V can…

A: According to the given information suppose,

Q: 2) Write the vector v as a sum of proj span{.…...} (v) and proj spo span{m} orthogonal set V₁,..., V…

A:

Q: Show that W is a vector space.

A:

Q: If V₁, V2 and v3 are 3 vectors in a vector space V such that -7v₁-9v2 = 0. V, and v₂ are linearly…

A: Given that v1, v2, and v3 are 3 vectors in a vector space Vsuch that -7v1-9v2=0. Then v1, v2,and v3…

Q: Prove or give a counterexample: If v1, V2, .. Vm and w1, W2, .. Wm are linearly independent lists of…

A:

Q: Are the vector spaces R4, M2,2, and M1,4 exactly the same? Describe their similarities and…

A: We have to determine if the vector spaces, are exactly the same or there is any similarity or…

Q: The shape's vector set is a subspace of R3

A: solution :-

Q: (b) (x1, Yi, Z1) + (X2, Y2 Z2) = (0, 0, 0) c(x, y, z) = (cx, cy, cz) O Thể set is a vector space. O…

A: b. A set S is said to be a vector space if for any vectors a,b in S and scalars s,t, as+bt belongs…

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 v1 , v2 , . . . , vn be linearly independent vectors in a vector space V. Show that v2, . . . ,…

A: If the given vectors are linearly independent in the vector space V. Then it follows that

Q: 6. Let V be a vector space and v1, v2, V3, V4 are vectors in V. Suppose that {v1, v2, V3} are linear…

A:

Q: (a) Show that the vectors v1 = (1, 2, 3, 4), v2 = (0, 1, 0, – 1), and v3 = (1, 3, 3, 3), form a…

A: (a) The given vectors are: v1=(1,2,3,4), v2=(0,1,0,-1), v3=(1,3,3,3)

Q: Let x 0 and y 0. (a) If xly, show that {x, y} is a linearly independ- ent set. (b) Extend the result…

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Q: Suppose y1 ( x), y2 ( x), y3 ( x) are three different functions of x. The vector space they span…

A:

Q: Let P2 be the vector space of all polynomials of degree ≤ 2 with coefficients in R, and S= {1 + 2x,…

A: The basis for the vector space P2 is 1, x, x2. We have to write the matrix form for the set S. We…

Q: Let V be a K-vector space and U1, U2, U3 three sub-vector spaces of V. Show that: dim(U1) + dim(U2)…

A: first of all we must know about the dimension of the vector space.what is the dimension of the…

Q: Show that the set of all pairs of real numbers (x, y) with the operations (X1, Y1) + (x2, Y2) = (x1…

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Q: 2 2) Find a number t such that -3 is not linearly independent in R.

A:

Q: 9. Determine whether the vectors v₁ = (1,1,2), v₂ = (1,0,1) and v3 = (2,1,3) span the vector space…

A: We know that the dimension of vector space  is 3.If the given three vectors are linearly…

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A:

Q: Let x1,x2,...,xk be linearly independent vectors in a vector space V. (a) If we add a vector xk+1…

A: The solution is given as

Q: 1. Let V₁, V2,...,Vn be n vectors in a vector space V. Explain what it means to say that these…

A:

Q: 3. Let V be the set of ordered pairs (a, b) of real numbers. Show that V is not a vector space over…

A: 3. Let  be the set of ordered pairs  of real numbers. Show that is not a vector space over  with…

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.