I'll ask you if some "simple" sets of vectors in (fairly) low-dimensions are linearly dependent (or independent). Here are some examples.

Example 1: Suppose u1, u2, and u3 are these vectors in R4 shown in order as follows:

[4]   [1 ]   [0]
[3] , [1 ] , [2] . 
[0]   [-1]   [0]
[0]   [0 ]   [5]
Is the set {u1, u2, u3} linearly independent?
An answer is here.

Example 2: Suppose u1, u2, and u3 are these vectors in R5 shown in order as follows:

[1]   [3]   [-5]
[2]   [4]   [-5]   
[3] , [5] , [-5] .
[4]   [6]   [-5]
[5]   [7]   [-5]
Is the set {u1, u2, u3} linearly independent?
An answer is here.

Example 3: Suppose u1, u2, u3, and u4 are these vectors in R4 shown in order as follows:

[431]   [231]   [2]   [-1]   
[ 2 ] , [ 6 ] , [4] , [-2] .
[260]   [111]   [2]   [-1]
[ 5 ]   [ 8 ]   [4]   [-2]
Is the set {u1, u2, u3, u4} linearly independent?
An answer is here.

Example 4: Suppose u1, u2, u3, and u4 are these vectors in R5 shown in order as follows:

[3]   [2]   [0]   [-3]   
[3]   [2]   [0]   [-1]   
[2] , [6] , [0] , [-2] .
[6]   [1]   [0]   [-1]
[5]   [8]   [0]   [-2]
Is the set {u1, u2, u3, u4} linearly independent?
An answer is here.

Example 5: Suppose u1, u2, u3, u4, and u5 are these vectors in R2 shown in order as follows:

[1] , [4] , [5] , [3] , [2] . 
[1]   [5]   [8]   [6]   [3]
Is the set {u1, u2, u3, u4, u5} linearly independent?
An answer is here.


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