Consider
With the operations
defined as.
Consider two vectors
Then we have
and
It is observed that
Therefore
is not commutative.
Hence
is not a vector space.
7. Show that V = R2 with the given operations and o is not a vector...
please answer the question
below
Show that the set R2, equipped with operations (x1, y1)F(x2, y2) = (x1 + x2 + 1, y1 + y2 – 1) A: (2, 3) = (Ag+1 – 1, 2g - A+1) defines a vector space over R. Show that the vector space V defined in question 1 is isomorphic to R² equipped with its usual vector space operations. This means you need to define an invertible linear map T:V R2.
Let V = R2 with the following operations: (zı, yı) + (2 2,32) = (x1 +T2-1, yı +B2) (addition) c(x1, y) = (czi-e+ 1, cy) where c E R (scalar multiplication). Then V is a vector space with these operations (you can take this as given). (a) (2) Let (-2,4) and (2,3) belong to V and let c -2 R. Find ca + y using the operations defined on V. (b) (2) What is the zero vector in V? Justify....
Let V = R3. Show that V with the given operations for and is not a vector space. Clearly explain what goes wrong in terms of at least one of the axioms for vector spaces. C1 C2 T1+ 2 +5 21 L222 15 and T1 cc1 21 CZ1
7. V={[)a620) a vector space! Draw the vector space? Draw the graph and explain why or why not? I. Verify the axiom for polynomial. p(x) = 2t' +31° +1+1 9(x) = 4r +57 +31 + 2 8. p(t)+9(1) € P. 9. p(t)+q(t) = f(t)+p(1) 10. cp(1) EP A subspace of a vector V is a subset H that satisfies what three conditions? 12. Is 0 a subspace of R" 13. Let V, V, E V; show H = span{v. v)...
linear algebra
1. Determine whether the given set, along with the specified operations of addition and scalar multiplication, is a vector space (over R). If it is not, list all of the axioms that fail to hold. a The set of all vectors in R2 of the form , with the usual vector addition and scalar multiplication b) R2 with the usual scalar multiplication but addition defined by 31+21 y1 y2 c) The set of all positive real numbers, with...
6. For each of the following, a subset W of a vector space V is given. Carefully prove or justify your answers. Use counterexamples where appropriate. a1 (b) (7 points) Show that w = | | a2 is a subspace of V-R4 under the usual operations. a4-аз-аг-ai a4
Show work
1. Assume you are given an mxn matrix H, an n-dimensional vector spacev h:VW are there such that H RepBp(h)? B, and an m-dimensional vector space Wwith basis D. How many linear mappings A) None B) One C) It depends on m and n. D) Infinitely many 2. Which of the following matrices will change from the basi the basis(31),(7))? R2 to -1'1 3 -2 B) 3 2 A) C) D) 3. Assume you are given an n-dimensional...
Linear Algebra
Exercise 7.24 Show that, for any line 1-span(v) in R2 and ally vector have R2, we Rex) (Rx)x.x i.e. reflections preserve the dot product). Conclude that reflections preserve lengths for all x eR2
Exercise 7.24 Show that, for any line 1-span(v) in R2 and ally vector have R2, we Rex) (Rx)x.x i.e. reflections preserve the dot product). Conclude that reflections preserve lengths for all x eR2
Why does this show that H is a subspace of R3? O A. The vector v spans both H and R3, making H a subspace of R3. OB. The span of any subset of R3 is equal to R3, which makes it a vector space. OC. It shows that H is closed under scalar multiplication, which is all that is required for a subset to be a vector space. OD. For any set of vectors in R3, the span of...
If addition and scalar multiplication is redefined on R2 in the following way, show it is not a vector space. (x1, yı) + (x2, y2) = (x1 + x2, Y1 + y2) and c(x, y) = (cx, y)