Determine if the columns of the matrix below form an indepen

Determine if the columns of the matrix below form an independent or dependent set.Use the formal definition of linear independence and row reduction to explain the answer.If the set is dependent, write one of the vectors as a linear combination of the others.

Solution

Linear Indepentence definition :

It means that if x, y, and z are each multiplied by a constant say a,b, and c, respectively, and set equal to zero, only the the trivial solution or in other words all constants are equal to zero will make this equation true.

that is ax+by+cz = 0

for this problem we\'ll be working with the columns of the given matrix.

Lets say x is the column [-4,0,1,2], y is the column [-3,-1,1,1], and z is the column [0,5,-5,-10]. 0 in this case would be in the form [0,0,0,0]

Now, multiply each column by its constant, and we\'ll get four equations

-4a -3b +0c = 0           -------------> (1)

0a -b + 5c = 0            --------------> (2)

a + b -5c = 0              --------------> (3)

2a + b - 10c = 0          -------------> (4)

from equation (2) and (3)

a = 0

now plug a = 0 in equations (1)

=> -3b = 0

or b = 0

now plug a = 0 and b = 0 in the eqution (4)

=> -10c = 0

or c = 0

So, going by the definition of linear independence, as it requires all constants {a , b abd c} to be zero in order for the equation ax + by + cz = 0,

Hence these columns are linearly independent