Lesson test: Linear combinations, span and independence

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Linear combinations, span and independence

1.[2p]
The span of two non-zero vectors in $R^{3}$ that are not multiples of each other is

a plane through the origin

Correct

a line through the origin

The answer is: a plane through the origin

all of $R^{3}$

The answer is: a plane through the origin

a plane that need not pass through the origin

The answer is: a plane through the origin

2.[1p]
Every span contains the zero vector.

True

Correct

False

The answer is: True

3.[2p]
Write $(4, 5, 6)$ as $a (1, 1, 1) + b (1, 2, 3)$ . What is $a$ ?

CorrectNot quite: 3

4.[2p]
A set of vectors is linearly dependent when

some combination with at least one non-zero coefficient gives the zero vector

Correct

every combination of them gives the zero vector

The answer is: some combination with at least one non-zero coefficient gives the zero vector

the combination with all coefficients zero gives the zero vector

The answer is: some combination with at least one non-zero coefficient gives the zero vector

none of them is the zero vector

The answer is: some combination with at least one non-zero coefficient gives the zero vector

5.[3p]
Which of these sets are linearly dependent?

Select all that apply

$(1, 1, 2)$ , $(2, 1, 3)$ , $(3, 2, 5)$

Correct

$(1, 2, 3)$ , $(2, 4, 6)$

Correct

$(1, 0, 0)$ , $(1, 1, 0)$ , $(1, 1, 1)$

The answer is: $(1, 1, 2)$, $(2, 1, 3)$, $(3, 2, 5)$, $(1, 2, 3)$, $(2, 4, 6)$

$(1, 0)$ , $(0, 1)$

The answer is: $(1, 1, 2)$, $(2, 1, 3)$, $(3, 2, 5)$, $(1, 2, 3)$, $(2, 4, 6)$

6.[2p]
Any four vectors in $R^{3}$ are linearly dependent.

True

Correct

False

The answer is: True

7.[3p]
Match each condition on the columns of a system to what it controls.

The target lies in the span of the columns
The columns are independent
The columns are dependent and the target is reachable
The target lies outside the span

infinitely many solutions
at least one solution exists
no solution
at most one solution exists

Show the answer

The target lies in the span of the columns: at least one solution exists The columns are independent: at most one solution exists The columns are dependent and the target is reachable: infinitely many solutions The target lies outside the span: no solution

8.[2p]
The columns of a system are $(1, 2)$ , $(2, 4)$ and $(3, 6)$ , and the target is $(5, 11)$ . How many solutions does it have?

CorrectNot quite: 0

9.[2p]
What is the smallest number of vectors that can span $R^{4}$ ?

CorrectNot quite: 4

1.[2p] The span of two non-zero vectors in R3 that are not multiples of each other is

2.[1p] Every span contains the zero vector.

3.[2p] Write (4,5,6) as a(1,1,1)+b(1,2,3). What is a?

4.[2p] A set of vectors is linearly dependent when

5.[3p] Which of these sets are linearly dependent?

6.[2p] Any four vectors in R3 are linearly dependent.

7.[3p] Match each condition on the columns of a system to what it controls.

8.[2p] The columns of a system are (1,2), (2,4) and (3,6), and the target is (5,11). How many solutions does it have?

9.[2p] What is the smallest number of vectors that can span R4?

1.[2p]
The span of two non-zero vectors in $R^{3}$ that are not multiples of each other is

2.[1p]
Every span contains the zero vector.

3.[2p]
Write $(4, 5, 6)$ as $a (1, 1, 1) + b (1, 2, 3)$ . What is $a$ ?

4.[2p]
A set of vectors is linearly dependent when

5.[3p]
Which of these sets are linearly dependent?

6.[2p]
Any four vectors in $R^{3}$ are linearly dependent.

7.[3p]
Match each condition on the columns of a system to what it controls.

8.[2p]
The columns of a system are $(1, 2)$ , $(2, 4)$ and $(3, 6)$ , and the target is $(5, 11)$ . How many solutions does it have?

9.[2p]
What is the smallest number of vectors that can span $R^{4}$ ?