In the previous segment, you understood the concept of basis vectors and how they’re the most fundamental units through which you explain the vectors. Now, let’s go ahead and understand how you can use different basis to explain the same set of vectors, similar to how you can use different units to explain the same measure.

As explained in the video above, using the analogy of basis as a unit of representation, different basis vectors can be used to represent the same observations, just like you can represent the weight of a patient in kilograms or pounds. As in the previous case, the basis vectors for the representation of the patient’s information is given by **[1ft0lbs] **and **[0ft1lbs]**.

The following table summarises the results you get when you make the change.

As you can see, the patient’s height and weight have not changed physically. It’s just that you’re using a different set of basis vectors now to explain the same patients. So **[16555]** is the same as **[5.4121.3]**when different basis vectors are being used.

**Relationship between the two sets of basis vectors**

To understand the relationship between the two basis vectors in concrete terms, recall the way we introduced it in the previous segment. We said that every vector in the 2D space can be written as a linear combination of the basis vectors.

So Patient 1’s information in the cm/kg space is given by 165⋅[10]+55⋅[01] whereas in the ft/lbs space is given by 5.4⋅[10]+121.3⋅[01]

Now, 1 ft = 30.48 cm and 1 cm = 0.033 ft

Similarly, 1 kg = 2.205 lbs and 1lbs = 0.454 kg.

Therefore, comparing the basis vectors, we can say

**[1ft0lbs]**?in ft/lbs space = **[30.48cm0kg]** in cm/kg space and

**[0ft1lbs]** in ft/lbs space = ?**[0cm0.45kg]**?

Here’s a neat manipulation that you can do to understand the way the numbers arrange amongst themselves using the linear combination property.

**[16555]** = **165[10]** + **55[01]** = **5.4[30.480]** +** 121.3[00.45]**in the cm/kg space.

In the above case, we considered the new basis vectors as **[30.480]** and **[00.45]** in the cm/kg space which is equivalent to (1,0) and (0,1) in the ft/lbs space. And using this, we got the representation of **[5.4121.3] **for the patient.

Therefore, we can choose a completely different set of vectors, say **v1** and **v2** as the basis vectors and find the representation of Patient 1 (originally in the standard basis vectors) in the new basis system. They should be satisfying the following linear combination equation

**[16555]**= a1⋅v1+a2⋅v2

where (a1,a2) is the representation of Patient 1 in the v1 and v2 space.

To understand better,

Taking v1=[30.480] and v2=[00.45] we got a1=5.4 and a2=121.3

Similarly, taking v1=[550] and v2=[055] we get a1=3 and a2=1

Again, taking v1=[31] and v2=[20] we get a1=55 and a2=0

and so on..

Did you notice something different in the above example where we considered v1=[31] and v2=[20]? This means that the new basis need not be parallel to the original basis.

Simply put, you have the flexibility of choosing a different set of basis vectors apart from the standard basis vectors that are provided to you to represent your information. The information won’t change, just the numbers representing the information would change.

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