Linear independence Definition ® The set {v 1, ..., vp} is said to be ...

LINEAR INDEPENDENCE [1.7] 

Q: Why do we care about linear independence? 

A: When expressing a vector x as a linear combination of 

a system {v1, · · · , vp} that is linearly dependent, then we 

can find a smaller system in which we can express x 

➤ A dependent system is ‘redundant’ 

 

✍ Let v1 = 

1 

1 

. Is {v1} linearly independent? [special 

case where p = 1 

✍ A system consisting of a nonzero vector [at least one 

nonzero entry] is always linearly independent: True - False? 

✍ Are the following systems linearly independent: 

 

1 

0 

, 

−1 

1 

, 

1 

0 

, 

−10 

0 

, 

1 

0 

, 

−1 

1 

, 

2 

1 

? 

F-3 Csci 2033 – September 25, 2012 

Linear independence 

Definition 

➤ The set {v1, ..., vp} is said to be linearly dependent if 

there exist weights c1, ..., cp, not all zero, such that 

c1v1 + c2v2 + ... + cpvp = 0 

➤ It is linearly independent otherwise 

➤ The above equation is called linear dependence relation 

among the vectors v1, · · · , vp 

➤ Another way to express dependence: A set of vectors is 

linearly dependent if and only if there is one vector among 

them which is a linear combination of all the others. 

✍ Prove this 


✍ Let 

⎡ ⎤ 

1 

⎢ ⎥ 

v1 = ⎣1⎦; 

⎡ ⎤ 

4 

⎢ ⎥ 

v2 = ⎣1⎦; 

⎡ ⎤ 

−2 

⎢ ⎥ 

v3 = ⎣ 3 ⎦; 

2 

5 

1 

(a) Determine if {v1, v2, v3} is linearly independent 

(b) If possible find a linear dependence relation among 

v1, v2, v3. 

Solution: We must determine if the system: 

⎡ ⎤ ⎡ ⎤ ⎡ ⎤ ⎡ ⎤ 

1 4 −2 0 

⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ 

x1 ⎣1⎦ 

+ x2 ⎣1⎦ 

+ x3 ⎣ 3 ⎦ = ⎣0⎦ 

2 5 1 0 

has a nontrivial solution 

Note Solution is trivial when x1 = x2 = x3 = 0 

F-4 Csci 2033 – September 25, 2012

Augmented syst: 

1 4 −2 0 

1 1 3 0 

2 5 1 0 

Echelon 1st step 

1 4 −2 0 

0 −3 5 0 

0 −3 5 0 

➤ This system is equivalent to original one. 

Echelon 2nd step 

1 4 −2 0 

0 −3 5 0 

0 0 0 0 

➤ Select x3 = 3 (to avoid fractions) and back-solve for x2 

(x2 = 5), and x1, (x1 = −14) 

➤ Conclusion: there is a nontrivial solution 

➤ NOT independent 

(b) Linear dependence relation: From above, 

−14v1 + 5v2 + v3 = 0 


LINEAR MAPPINGS [1.8] 

Note: Text uses the reduced echelon form instead of backsolving 

[result is clearly the same. Both solution are OK] 

➤ With the reduced row echelon form 

1 0 14/3 0 

0 1 −5/3 0 

0 0 0 0 

➤ x1 = −(14/3)x3; x2 = (5/3)x3 

➤ select x3 = 3 then x2 = 5, x1 = 14 

➤ Recall: x1, x2 are basic variables, and x3 is free 


Introduction to linear mappings [1.8] 

➤ A transformation or function or mapping from R n to R m 

is a rule which assigns to every x in R n a vector T (x) in 

R m . 

➤ R n is called the domain 

space of T and R m the 

image space or co-domain 

of T . 

➤ Notation: 

T : R n −→ R m 

R n 

● 

x 

Domain 

➤ T (x) is the image of x under T 

T 

R m 

● 

Range 

T(x) 

Image / Codomain 


Example: Take the mapping from R 2 to R 3 : 

T : R 2 

x = 

 

x1 

x2 

 

−→ R 3 

⎡ 

⎢ 

−→T (x) = ⎣ 

x1 + x2 

x1x2 

x 2 1 + x2 2 

Example: Another mapping from R 2 to R 3 : 

T : R 2 

x = 

 

x1 

x2 

 

−→ R 3 

−→T (x) = 

⎡ 

⎢ 

⎣ 

x1 + x2 

x1 − x2 

x1 + 5x2 

✍ What is the main difference between these 2 examples? 


➤ Given an m × n matrix A Consider the special mapping: 

T :R n −→R m 

x −→y = Ax 

✍ Domain == ??; Image space == ?? 

➤ From what we saw earlier [‘Properties of the matrixvector 

product’] such mappings are linear 

➤ As it turns out: 

Any linear mapping can be represented in matrix form 

➤ This means that if T is linear, there exists a matrix A 

such that T (x) = Ax for all x in R n . 

➤ A is the representation of T . 


⎤ 

⎥ 

⎦ 

⎤ 

⎥ 

⎦ 

Definition A mapping T is linear if: 

(i) T (u + v) = T (u) + T (v) for u, v in the domain of T 

(ii) T (αv) = αT (u) for all scalars α and all u in the 

domain of T 

➤ The mapping of the second example given above is linear 

- but not for the first one. 

➤ If a mapping is linear then T (0) = 0 

Observation: A mapping is linear if and only if 

T (αu + βv) = αT (u) + βT (v) 

for all scalars α, β and all u, v in the domain of T . 


✍ Let A be a square matrix. Is the mapping x → x + Ax 

linear? If so find the matrix associated with it. 

✍ Same questions for the mapping x → Ax + αx - where 

α is a scalar. 

✍ Express the following mapping from 

R 3 to R 2 in matrix/vector form: 

➤ Is this a linear mapping? 

y1 = 2x1 − x2 +1 

y2 = x2 − x3 −2 

✍ Read Section 1.9 and explore the notions of onto mappings 

(‘surjective’) and one-to-one mappings (‘injective’) 

in the text. You must at least know the definitions. 

✍ A mapping is onto if and only if .... 

✍ A mapping is one-to-one if and only if ....

Linear independence Definition ® The set {v 1, ..., vp} is said to be ...

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