Lesson test: Subspaces, basis and dimension

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Subspaces, basis and dimension

1.[3p]
Which of these are subspaces of $R^{2}$ ?

Select all that apply

The line through the origin in the direction $(1, 2)$

Correct

The set containing only the zero vector

Correct

The line $y = 2 x + 1$

The answer is: The line through the origin in the direction $(1, 2)$, The set containing only the zero vector

The first quadrant, where $x \geq 0$ and $y \geq 0$

The answer is: The line through the origin in the direction $(1, 2)$, The set containing only the zero vector

2.[2p]
The first quadrant fails to be a subspace because

it is not closed under multiplication by a negative scalar

Correct

it does not contain the zero vector

The answer is: it is not closed under multiplication by a negative scalar

it is not closed under addition

The answer is: it is not closed under multiplication by a negative scalar

it is unbounded

The answer is: it is not closed under multiplication by a negative scalar

3.[2p]
A basis of a subspace is a set that is

independent and spanning

Correct

spanning, with as many vectors as possible

The answer is: independent and spanning

independent, and contained in the standard basis

The answer is: independent and spanning

orthogonal and of length one

The answer is: independent and spanning

4.[2p]
What is the dimension of the plane $x + 2 y - z = 0$ inside $R^{3}$ ?

CorrectNot quite: 2

5.[2p]
Coordinates with respect to a given basis are unique, and independence is what makes them so.

True

Correct

False

The answer is: True

6.[2p]
Express $(7, 4)$ as $a (1, 1) + b (1, - 1)$ . What is $a$ ?

CorrectNot quite: 5.5

7.[2p]
What is the dimension of the space of polynomials of degree at most five?

CorrectNot quite: 6

8.[3p]
Match each space to its dimension.

A line through the origin
The subspace containing only the zero vector
All of $R^{4}$
The solutions of $y^{''} + y = 0$

Show the answer

A line through the origin: 1 The subspace containing only the zero vector: 0 All of $R^{4}$ : 4 The solutions of $y^{''} + y = 0$ : 2

9.[3p]
In a space of dimension $d$ , a set of exactly $d$ independent vectors

automatically spans the space

Correct

may still fail to span the space

The answer is: automatically spans the space

must be the standard basis

The answer is: automatically spans the space

must be orthogonal

The answer is: automatically spans the space

1.[3p] Which of these are subspaces of R2?

2.[2p] The first quadrant fails to be a subspace because

3.[2p] A basis of a subspace is a set that is

4.[2p] What is the dimension of the plane x+2y-z=0 inside R3?

5.[2p] Coordinates with respect to a given basis are unique, and independence is what makes them so.

6.[2p] Express (7,4) as a(1,1)+b(1,-1). What is a?

7.[2p] What is the dimension of the space of polynomials of degree at most five?

8.[3p] Match each space to its dimension.

9.[3p] In a space of dimension d, a set of exactly d independent vectors

1.[3p]
Which of these are subspaces of $R^{2}$ ?

2.[2p]
The first quadrant fails to be a subspace because

3.[2p]
A basis of a subspace is a set that is

4.[2p]
What is the dimension of the plane $x + 2 y - z = 0$ inside $R^{3}$ ?

5.[2p]
Coordinates with respect to a given basis are unique, and independence is what makes them so.

6.[2p]
Express $(7, 4)$ as $a (1, 1) + b (1, - 1)$ . What is $a$ ?

7.[2p]
What is the dimension of the space of polynomials of degree at most five?

8.[3p]
Match each space to its dimension.

9.[3p]
In a space of dimension $d$ , a set of exactly $d$ independent vectors