Worksheet 2
Problems using RREF.
These are problems are all variations on a theme, using Reduced Row Echelon Form.
restart;
with(LinearAlgebra): with(Student[LinearAlgebra]):
Problem 1.
Consider the matrix
A := Matrix([[2, -3, 4, -8, -17, -10], [1, -1, 1, -4, -8, -6], [6, -3, 0, -10, -3, -14], [-7, 0, 7, 13, 4, 26], [-1, 1, -1, 0, -4, -2]]);
Part A.
Find a basis for the Nullspace of A.
Part B.
Find a basis of the Row Space of A.
Part C.
Find a basis of the Column Space of A. What is the Rank of A?
Part D.
Ask maple the same questions with the commands NullSpace, RowSpace, ColumnSpace and Rank. Do your results agree?
Problem 2.
In each part, determine if the given vectors are linearly independent or linearly dependent. If they are dependent, find scalars 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 all zero, so that 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.
Part A.
v[1]:=Vector(4, {(1) = 41, (2) = -8, (3) = 29, (4) = -8}); v[2]:=Vector(4, {(1) = -13, (2) = 3, (3) = -5, (4) = 3}); v[3]:=Vector(4, {(1) = -2, (2) = 0, (3) = -5, (4) = 0});
Part B.
Using a shorthand way of entering the vectors.
(v[1], v[2], v[3]):=Vector(4, {(1) = 1, (2) = 1, (3) = -2, (4) = 1}), Vector(4, {(1) = -1, (2) = 0, (3) = 7, (4) = -3}), Vector(4, {(1) = -8, (2) = -3, (3) = 41, (4) = -18});
v[1],v[2],v[3];
Problem 3.
In each part, determine if the list of vectors spans 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If so, extract a basis from the list.
Part A.
(v[1],v[2],v[3],v[4],v[5]):=Vector(4, {(1) = -1, (2) = -1, (3) = 1, (4) = 6}), Vector(4, {(1) = 2, (2) = 3, (3) = -2, (4) = -11}), Vector(4, {(1) = 1, (2) = -3, (3) = 0, (4) = -7}), Vector(4, {(1) = -3, (2) = 6, (3) = 1, (4) = 21}), Vector(4, {(1) = 7, (2) = 1, (3) = -5, (4) = -42});
Part B.
(v[1],v[2],v[3],v[4],v[5]) := Vector(4, {(1) = -178, (2) = -73, (3) = -199, (4) = 38}), Vector(4, {(1) = -182, (2) = -75, (3) = -204, (4) = 39}), Vector(4, {(1) = -114, (2) = -47, (3) = -126, (4) = 24}), Vector(4, {(1) = 19, (2) = 8, (3) = 21, (4) = -4}), Vector(4, {(1) = -627, (2) = -258, (3) = -697, (4) = 133});
s
Problem 4.
The following vectors are linearly independent. Complete the list to a basis of LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUklbXN1cEdGJDYlLUkjbWlHRiQ2J1EiUkYnLyUlYm9sZEdRJXRydWVGJy8lJ2l0YWxpY0dGNC8lLG1hdGh2YXJpYW50R1EsYm9sZC1pdGFsaWNGJy8lK2ZvbnR3ZWlnaHRHUSVib2xkRictRiM2JC1JI21uR0YkNiRRIjVGJy9GOFEnbm9ybWFsRidGQy8lMXN1cGVyc2NyaXB0c2hpZnRHUSIwRidGQw== by adding on some of the standard basis vectors.
(v[1],v[2],v[3]):=Vector(5, {(1) = 1, (2) = 1, (3) = 2, (4) = -2, (5) = 0}), Vector(5, {(1) = 0, (2) = 0, (3) = -5, (4) = 2, (5) = 1}), Vector(5, {(1) = -10, (2) = -5, (3) = 22, (4) = -7, (5) = -5});
Problem 5.
Consider the vectors
(v[1],v[2],v[3],v[4],v[5],v[6]):=Vector(4, {(1) = -1, (2) = -1, (3) = 1, (4) = 6}), Vector(4, {(1) = 2, (2) = 2, (3) = -2, (4) = -12}), Vector(4, {(1) = 2, (2) = 3, (3) = -2, (4) = -11}), Vector(4, {(1) = 13, (2) = 18, (3) = -13, (4) = -73}), Vector(4, {(1) = 1, (2) = -3, (3) = 0, (4) = -7}), Vector(4, {(1) = 9, (2) = -2, (3) = -6, (4) = -56});
(w[1],w[2],w[3]):= Vector(4, {(1) = 4, (2) = -3, (3) = -2, (4) = -25}), Vector(4, {(1) = 6, (2) = 4, (3) = -5, (4) = -35}), Vector(4, {(1) = 5, (2) = 3, (3) = -4, (4) = -30});
Let LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEiU0YnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic= be the subspace 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.
Part A.
Find a basis for LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEiU0YnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic=. What is the dimension of LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEiU0YnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic=?
Part B.
For each of the vectors 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, determine if the vector is in LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEiU0YnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic= and, if so, express it as a linear combination of the basis vectors for LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEiU0YnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic= you found in the first part of the problem.
Problem 6.
In each part, let LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUklbXN1YkdGJDYlLUkjbWlHRiQ2JVEiU0YnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1GIzYkLUkjbW5HRiQ2JFEiMUYnL0Y2USdub3JtYWxGJ0Y+LyUvc3Vic2NyaXB0c2hpZnRHUSIwRidGPg== be the subspace spanned by the vectors LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUklbXN1YkdGJDYlLUkjbWlHRiQ2J1EidkYnLyUlYm9sZEdRJXRydWVGJy8lJ2l0YWxpY0dGNC8lLG1hdGh2YXJpYW50R1EsYm9sZC1pdGFsaWNGJy8lK2ZvbnR3ZWlnaHRHUSVib2xkRictRiM2JC1GLzYlUSJpRidGNS9GOFEnaXRhbGljRicvRjhRJ25vcm1hbEYnLyUvc3Vic2NyaXB0c2hpZnRHUSIwRidGRA== and let LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUklbXN1YkdGJDYlLUkjbWlHRiQ2JVEiU0YnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1GIzYkLUkjbW5HRiQ2JFEiMkYnL0Y2USdub3JtYWxGJ0Y+LyUvc3Vic2NyaXB0c2hpZnRHUSIwRidGPg== be the subspace spanned by the vectors 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 Answer the following questions
Is LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUklbXN1YkdGJDYlLUkjbWlHRiQ2JVEiU0YnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1GIzYkLUkjbW5HRiQ2JFEiMUYnL0Y2USdub3JtYWxGJ0Y+LyUvc3Vic2NyaXB0c2hpZnRHUSIwRidGPg== contained in LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUklbXN1YkdGJDYlLUkjbWlHRiQ2JVEiU0YnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1GIzYkLUkjbW5HRiQ2JFEiMkYnL0Y2USdub3JtYWxGJ0Y+LyUvc3Vic2NyaXB0c2hpZnRHUSIwRidGPg==?
Is LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUklbXN1YkdGJDYlLUkjbWlHRiQ2JVEiU0YnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1GIzYkLUkjbW5HRiQ2JFEiMkYnL0Y2USdub3JtYWxGJ0Y+LyUvc3Vic2NyaXB0c2hpZnRHUSIwRidGPg== contained in LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUklbXN1YkdGJDYlLUkjbWlHRiQ2JVEiU0YnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1GIzYkLUkjbW5HRiQ2JFEiMUYnL0Y2USdub3JtYWxGJ0Y+LyUvc3Vic2NyaXB0c2hpZnRHUSIwRidGPg==?
Does LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYnLUklbXN1YkdGJDYlLUkjbWlHRiQ2JVEiU0YnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1GIzYkLUkjbW5HRiQ2JFEiMUYnL0Y2USdub3JtYWxGJ0Y+LyUvc3Vic2NyaXB0c2hpZnRHUSIwRictSSNtb0dGJDYtUSI9RidGPi8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGSS8lKXN0cmV0Y2h5R0ZJLyUqc3ltbWV0cmljR0ZJLyUobGFyZ2VvcEdGSS8lLm1vdmFibGVsaW1pdHNHRkkvJSdhY2NlbnRHRkkvJSdsc3BhY2VHUSwwLjI3Nzc3NzhlbUYnLyUncnNwYWNlR0ZYLUZENi1RIn5GJ0Y+RkdGSkZMRk5GUEZSRlQvRldRJjAuMGVtRicvRlpGaW4tRiw2JUYuLUYjNiQtRjs2JFEiMkYnRj5GPkZARj4=?
Part A.
(v[1],v[2],v[3],v[4]):=Vector(5, {(1) = -27, (2) = -27, (3) = 180, (4) = 57, (5) = -26}), Vector(5, {(1) = 64, (2) = 63, (3) = -422, (4) = -133, (5) = 60}), Vector(5, {(1) = -17, (2) = -17, (3) = 113, (4) = 36, (5) = -17}), Vector(5, {(1) = 67, (2) = 66, (3) = -443, (4) = -139, (5) = 61});
(w[1],w[2],w[3],w[4]):=Vector(5, {(1) = 319, (2) = 314, (3) = -2106, (4) = -662, (5) = 294}), Vector(5, {(1) = 70, (2) = 68, (3) = -459, (4) = -143, (5) = 61}), Vector(5, {(1) = -5, (2) = -4, (3) = 24, (4) = 10, (5) = -12}), Vector(5, {(1) = -30, (2) = -30, (3) = 201, (4) = 63, (5) = -27});
Part B.
(v[1],v[2],v[3],v[4]):=Vector(5, {(1) = 13, (2) = -1, (3) = 7, (4) = 1, (5) = -1}), Vector(5, {(1) = 2, (2) = 0, (3) = 1, (4) = 0, (5) = 0}), Vector(5, {(1) = -6, (2) = 0, (3) = -3, (4) = 0, (5) = 1}), Vector(5, {(1) = -42, (2) = 2, (3) = -22, (4) = -2, (5) = 5});
(w[1], w[2], w[3], w[4]):=Vector(5, {(1) = 17, (2) = -1, (3) = 9, (4) = 1, (5) = -1}), Vector(5, {(1) = -29, (2) = 3, (3) = -16, (4) = -3, (5) = 3}), Vector(5, {(1) = 17, (2) = -1, (3) = 9, (4) = 1, (5) = -1}), Vector(5, {(1) = -37, (2) = 3, (3) = -20, (4) = -3, (5) = 3});
Part C.
(v[1], v[2], v[3], v[4]):=Vector(5, {(1) = -2, (2) = 1, (3) = -1, (4) = 0, (5) = -4}), Vector(5, {(1) = 27, (2) = -9, (3) = 11, (4) = 0, (5) = 58}), Vector(5, {(1) = 5, (2) = -2, (3) = 2, (4) = -1, (5) = 11}), Vector(5, {(1) = 71, (2) = -25, (3) = 29, (4) = -3, (5) = 153});
(w[1], w[2], w[3], w[4]):=Vector(5, {(1) = -1, (2) = -5, (3) = 1, (4) = 0, (5) = -1}), Vector(5, {(1) = -3, (2) = -6, (3) = 2, (4) = 1, (5) = -2}), Vector(5, {(1) = -2, (2) = 7, (3) = 0, (4) = 2, (5) = -2}), Vector(5, {(1) = -4, (2) = -94, (3) = 13, (4) = -9, (5) = 1});