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İ S. Y

F ,

Surname, Name:

Q . §.V E

Is b=

1

−1

3

a linear combination of the vectors a1=

0

1

1

,a2=

1

1

2

and a3=

−1

−1

−2

?

A .

To answer this question, we need to solve the vector equation x1a1+x2a2+

x3a3=b. In other words, we need to reduce the augmented matrix [a1a2a3b].

0 1 −1 1

1 1 −1−1

1 2 −2−3

R1↔R2

//

1 1 −1−1

0 1 −1 1

1 2 −2−3

−R1+R3↔R3

//

1 1 −1−1

0 1 −1 1

0 1 −1−2

−R2+R3↔R3

//

1 1 −1−1

0 1 −1 1

0 0 0 −3

Note that the last augmented matrix corresponds to an inconsistent system.

Therefore, the vector equation x1a1+x2a2+x3a3=bhas no solution which

means that bis not a linear combination of a1,a2and a3.

Q . §. V E

Let v1=

1

0

−2

,v2=

−3

1

8

, and y=

h

−5

−3

. For what value(s) of his y

in the plane generated by v1and v2?

A .

To answer this question, we need to solve the vector equation x1v1+x2v2=y.

In other words, we need to reduce the augmented matrix [v1v2y].

1−3h

0 1 −5

−2 8 −3

2R1+R3↔R3

//

1−3h

0 1 −5

0 2 −3 + 2h

−2R2+R3↔R3

//

1−3h

0 1 −5

0 0 7 + 2h

Since we want y∈Span{v1,v2}, we need a consistent system. Therefore, we

have to have 7 + 2h= 0. In other words, yis in the plane generated by v1and

v2when h=−7/2.

1

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