Linear algebra - basis multiple choice questions

In summary, in the first conversation, the correct answer is d. In the second conversation, the correct answers are b, a, a, d, and a respectively. In the third conversation, the correct answers are a and b respectively.
  • #1
underacheiver
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0

Homework Statement


1. Which of the following is not a linear transformation from 3 to 3?
a. T(x, y, z) = (x, 2y, 3x - y)
b. T(x, y, z) = (x - y, 0, y - z)
c. T(x, y, z) = (0, 0, 0)
d. T(x, y, z) = (1, x, z)
e. T(x, y, z) = (2x, 2y, 5z)

2. Which of the following statements is not true?
a. If A is any n × m matrix, then the transformation T: defined by T(x) = Ax is always a linear transformation.
b. If T: U → V is any linear transformation from U to V then T(xy) = T(x)T(y) for all vectors x and y in U.
c. If T: U → V is any linear transformation from U to V then T(-x) = -T(x) for all vectors x in U.
d. If T: U → V is any linear transformation from U to V then T(0) = 0 in V for 0 in U.
e. If T: U → V is any linear transformation from U to V then T(2x) = 2T(x) for all vectors x in U.

3. If T: U → V is any linear transformation from U to V then
a. the kernel of T is a subspace of U
b. the kernel of T is a subspace of V
c. the range of T is a subspace of U
d. V is always the range of T
e. V is the range of T if, and only if, ket T = {0}

4. If T: U → V is any linear transformation from U to V and B = {u 1, u 2, ..., u n} is a basis for U, then set T(B) = {T(u 1), T(u 2), ... T(u n)}
a. spans V
b. spans U
c. is a basis for V
d. is linearly independent
e. spans the range of T

5. P 3 is a vector space of polynomials in x of degree three or less and Dx(p(x)) = the derivative of p(x) is a transformation from P 3 to P 2.
a. the nullity of Dx is two.
b. The polynomial 2x + 1 is in the kernel of Dx.
c. The polynomial 2x + 1 is in the range of Dx.
d. The kernel of Dx is all those polynomials in P 3 with zero constant term.
e. The rank of Dx is three.

6.Let Ax = b be the matrix representation of a system of equations. The system has a solution if, and only if, b is in the row space of the matrix A.
a. True
b. False

7.If A is an n × n matrix, then the rank of A equals the number of linearly independent row vectors in A.
a. True
b. False

Homework Equations



The Attempt at a Solution


1. d
2. b
3. a
4. a
5. d
6. b
7. a
 
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  • #2
4 and 5 are wrong
 
  • #3
How do you figure out the right answer??
4. if it doesn;t span U then does it span V?
5. is the answer b? because i don't see how it can be anything else then.
 
  • #4
Play with examples. For #4, use U=the plane, V=the plane, and B={i,j} (the standard basis). Can you give an example of T that isn't one-to-one? Now test each of the five responses with this T. I bet you can rule out four of them.

For #5, write down an actual cubic, and then find Dx of your cubic. Try another one. Pretty soon, you'll find that one of the five responses is obviously correct (and hopefully the other four are therefore wrong).
 
  • #5
i finally got b for 4 and e for 5
for 4, b completes the theorum in one of my textbooks.
for 5, i solved the nullity of Dx as 0, thus the rank has to be 3? is that correct?
 

1. What is a basis in linear algebra?

A basis in linear algebra is a set of linearly independent vectors that span a vector space. This means that any vector in the vector space can be expressed as a linear combination of the basis vectors.

2. How do you determine if a set of vectors form a basis?

To determine if a set of vectors form a basis, you can use the method of Gaussian elimination to reduce the matrix formed by the vectors to reduced row echelon form. If the resulting matrix has a pivot in every column, then the vectors form a basis. Alternatively, you can also check if the vectors are linearly independent and span the vector space.

3. Can a vector space have more than one basis?

Yes, a vector space can have multiple bases. For example, in two-dimensional space, the standard basis can be {(1,0), (0,1)}, but another possible basis could be {(2,0), (-1,3)}.

4. What is the difference between a basis and a spanning set?

A basis is a special type of spanning set that is both linearly independent and spans the vector space. A spanning set, on the other hand, may contain linearly dependent vectors and may not be the most efficient way to represent a vector space.

5. How is the dimension of a vector space related to its basis?

The dimension of a vector space is equal to the number of vectors in its basis. For example, a two-dimensional vector space will have a basis with two vectors, while a three-dimensional vector space will have a basis with three vectors.

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