1. Which of the following sets of vectors spans [r^2] ?

a) {(1,1), (-2,-2)}

b) {(1,1), (1,2)}

c) {(1,2), (1/2,2)}

d) {(-1,1), (1,-1)}



2.

Which of the following sets of vectors spans [r^3] ?

a) {(1,1,1), (2,2,2)}

b) {(1,3,1), (2,2,2)}

c) {(1,2,1), (1/2,1,1/2)}

d) {(1,3,2), (-1,-3,-2)}

1. Looks like (b) and (c) unless there's a typo maybe c = {(1,2),(1/2,1)}a as in #2?

2. (b) again

To determine whether a given set of vectors spans a particular vector space, you need to check if any vector in that space can be expressed as a linear combination of the given vectors.

For example, in question 1, we need to determine which set of vectors spans ℝ².

a) {(1,1), (-2,-2)}:
To determine if this set spans ℝ², we need to see if any vector in ℝ² can be written as a linear combination of (1,1) and (-2,-2). We can write the general form of a vector in ℝ² as (x, y). Let's try to express this vector as a linear combination of the given vectors: (1,1) * a + (-2,-2) * b = (x, y). Simplifying this equation gives the system of equations: a - 2b = x and a - 2b = y. This system of equations is inconsistent because the second equation is a multiple of the first equation. Hence, the set {(1,1), (-2,-2)} does not span ℝ².

b) {(1,1), (1,2)}:
Similarly, we can try to express any vector in ℝ² as a linear combination of (1,1) and (1,2). Let's write (x, y) as a linear combination: (1,1) * a + (1,2) * b = (x, y). Simplifying this equation gives the system of equations: a + b = x and a + 2b = y. This is a consistent system of equations, and we can solve it to express any vector in ℝ² as a linear combination of (1,1) and (1,2). Therefore, the set {(1,1), (1,2)} spans ℝ².

c) {(1,2), (1/2,2)}:
Similarly, we can write (x, y) as a linear combination of (1,2) and (1/2,2): (1,2) * a + (1/2,2) * b = (x, y). Simplifying this equation gives the system of equations: a + (1/2)b = x and 2a + 2b = y. This system of equations is consistent, and we can solve it to express any vector in ℝ² as a linear combination of (1,2) and (1/2,2). Therefore, the set {(1,2), (1/2,2)} spans ℝ².

d) {(-1,1), (1,-1)}:
In a similar manner, we can try to express any vector in ℝ² as a linear combination of (-1,1) and (1,-1): (-1,1) * a + (1,-1) * b = (x, y). Simplifying this equation gives the system of equations: -a + b = x and a - b = y. This system of equations is consistent, and we can solve it to express any vector in ℝ² as a linear combination of (-1,1) and (1,-1). Therefore, the set {(-1,1), (1,-1)} spans ℝ².

Hence, the sets that span ℝ² are {(1,1), (1,2)}, {(1,2), (1/2,2)}, and {(-1,1), (1,-1)}.

For question 2, you can follow the same process to determine which sets of vectors span ℝ³.